A004070 -id:A004070 - cp's OEIS frontend

A001045 Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer to 2^n/3.

0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, 178956971, 357913941, 715827883, 1431655765, 2863311531, 5726623061, 11453246123

Offset: 0

N. J. A. Sloane

Don Knuth points out (personal communication) that Jacobsthal may never have seen the actual values of this sequence. However, Horadam uses the name "Jacobsthal sequence", such an important sequence needs a name, and there is a law that says the name for something should never be that of its discoverer. - N. J. A. Sloane, Dec 26 2020
Number of ways to tile a 3 X (n-1) rectangle with 1 X 1 and 2 X 2 square tiles.
Also, number of ways to tile a 2 X (n-1) rectangle with 1 X 2 dominoes and 2 X 2 squares. - Toby Gottfried, Nov 02 2008
Also a(n) counts each of the following four things: n-ary quasigroups of order 3 with automorphism group of order 3, n-ary quasigroups of order 3 with automorphism group of order 6, (n-1)-ary quasigroups of order 3 with automorphism group of order 2 and (n-2)-ary quasigroups of order 3. See the McKay-Wanless (2008) paper. - Ian Wanless, Apr 28 2008
Also the number of ways to tie a necktie using n + 2 turns. So three turns make an "oriental", four make a "four in hand" and for 5 turns there are 3 methods: "Kelvin", "Nicky" and "Pratt". The formula also arises from a special random walk on a triangular grid with side conditions (see Fink and Mao, 1999). - arne.ring(AT)epost.de, Mar 18 2001
Also the number of compositions of n + 1 ending with an odd part (a(2) = 3 because 3, 21, 111 are the only compositions of 3 ending with an odd part). Also the number of compositions of n + 2 ending with an even part (a(2) = 3 because 4, 22, 112 are the only compositions of 4 ending with an even part). - Emeric Deutsch, May 08 2001
Arises in study of sorting by merge insertions and in analysis of a method for computing GCDs - see Knuth reference.
Number of perfect matchings of a 2 X n grid upon replacing unit squares with tetrahedra (C_4 to K_4):
o----o----o----o...
| \/ | \/ | \/ |
| /\ | /\ | /\ |
o----o----o----o... - Roberto E. Martinez II, Jan 07 2002
Also the numerators of the reduced fractions in the alternating sum 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 + ... - Joshua Zucker, Feb 07 2002
Also, if A(n), B(n), C(n) are the angles of the n-orthic triangle of ABC then A(1) = Pi - 2*A, A(n) = s(n)*Pi + (-2)^n*A where s(n) = (-1)^(n-1) * a(n) [1-orthic triangle = the orthic triangle of ABC, n-orthic triangle = the orthic triangle of the (n-1)-orthic triangle]. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Jun 05 2002
Also the number of words of length n+1 in the two letters s and t that reduce to the identity 1 by using the relations sss = 1, tt = 1 and stst = 1. The generators s and t and the three stated relations generate the group S3. - John W. Layman, Jun 14 2002
Sums of pairs of consecutive terms give all powers of 2 in increasing order. - Amarnath Murthy, Aug 15 2002
Excess clockwise moves (over counterclockwise) needed to move a tower of size n to the clockwise peg is -(-1)^n*(2^n - (-1)^n)/3; a(n) is its unsigned version. - Wouter Meeussen, Sep 01 2002
Also the absolute value of the number represented in base -2 by the string of n 1's, the negabinary repunit. The Mersenne numbers (A000225 and its subsequences) are the binary repunits. - Rick L. Shepherd, Sep 16 2002
Note that 3*a(n) + (-1)^n = 2^n is significant for Pascal's triangle A007318. It arises from a Jacobsthal decomposition of Pascal's triangle illustrated by 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 = (7 + 35 + 1) + (1 + 35 + 7) + (21 + 21) = 43 + 43 + 42 = 3a(7) - 1; 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = (1 + 56 + 28) + (28 + 56 + 1) + (8 + 70 + 8) = 85 + 85 + 86 = 3a(8)+1. - Paul Barry, Feb 20 2003
Number of positive integers requiring exactly n signed bits in the nonadjacent form representation.
Equivalently, number of length-(n-1) words with letters {0, 1, 2} where no two consecutive letters are nonzero, see example and fxtbook link. - Joerg Arndt, Nov 10 2012
Counts walks between adjacent vertices of a triangle. - Paul Barry, Nov 17 2003
Every amphichiral rational knot written in Conway notation is a palindromic sequence of numbers, not beginning or ending with 1. For example, for 4 <= n <= 12, the amphichiral rational knots are: 2 2, 2 1 1 2, 4 4, 3 1 1 3, 2 2 2 2, 4 1 1 4, 3 1 1 1 1 3, 2 3 3 2, 2 1 2 2 1 2, 2 1 1 1 1 1 1 2, 6 6, 5 1 1 5, 4 2 2 4, 3 3 3 3, 2 4 4 2, 3 2 1 1 2 3, 3 1 2 2 1 3, 2 2 2 2 2 2, 2 2 1 1 1 1 2 2, 2 1 2 1 1 2 1 2, 2 1 1 1 1 1 1 1 1 2. For the number of amphichiral rational knots for n=2*k (k=1, 2, 3, ...), we obtain the sequence 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, ... - Slavik Jablan, Dec 26 2003
a(n+2) counts the binary sequences of total length n made up of codewords from C = {0, 10, 11}. - Paul Barry, Jan 23 2004
Number of permutations with no fixed points avoiding 231 and 132.
The n-th entry (n > 1) of the sequence is equal to the 2,2-entry of the n-th power of the unnormalized 4 X 4 Haar matrix: [1 1 1 0 / 1 1 -1 0 / 1 1 0 1 / 1 1 0 -1]. - Simone Severini, Oct 27 2004
a(n) is the number of Motzkin (n+1)-sequences whose flatsteps all occur at level 1 and whose height is less than or equal to 2. For example, a(4) = 5 counts UDUFD, UFDUD, UFFFD, UFUDD, UUDFD. - David Callan, Dec 09 2004
a(n+1) gives row sums of A059260. - Paul Barry, Jan 26 2005
If (m + n) is odd, then 3*(a(m) + a(n)) is always of the form a^2 + 2*b^2, where a and b both equal powers of 2; consequently every factor of (a(m) + a(n)) is always of the form a^2 + 2*b^2. - Matthew Vandermast, Jul 12 2003
Number of "0,0" in f_{n+1}, where f_0 = "1" and f_{n+1} = a sequence formed by changing all "1"s in f_n to "1,0" and all "0"s in f_n to "0,1". - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
All prime Jacobsthal numbers A049883[n] = {3, 5, 11, 43, 683, 2731, 43691, ...} have prime indices except for a(4) = 5. All prime Jacobsthal numbers with prime indices (all but a(4) = 5) are of the form (2^p + 1)/3 - the Wagstaff primes A000979[n]. Indices of prime Jacobsthal numbers are listed in A107036[n] = {3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, ...}. For n>1 A107036[n] = A000978[n] Numbers n such that (2^n + 1)/3 is prime. - Alexander Adamchuk, Oct 03 2006
Correspondence: a(n) = b(n)*2^(n-1), where b(n) is the sequence of the arithmetic means of previous two terms defined by b(n) = 1/2*(b(n-1) + b(n-2)) with initial values b(0) = 0, b(1) = 1; the g.f. for b(n) is B(x) := x/(1-(x^1+x^2)/2), so the g.f. A(x) for a(n) satisfies A(x) = B(2*x)/2. Because b(n) converges to the limit lim (1-x)*B(x) = 1/3*(b(0) + 2*b(1)) = 2/3 (for x --> 1), it follows that a(n)/2^(n-1) also converges to 2/3 (see also A103770). - Hieronymus Fischer, Feb 04 2006
Inverse: floor(log_2(a(n))) = n - 2 for n >= 2. Also: log_2(a(n) + a(n-1)) = n - 1 for n >= 1 (see also A130249). Characterization: x is a Jacobsthal number if and only if there is a power of 4 (= c) such that x is a root of p(x) = 9*x*(x-c) + (c-1)*(2*c+1) (see also the indicator sequence A105348). - Hieronymus Fischer, May 17 2007
This sequence counts the odd coefficients in the expansion of (1 + x + x^2)^(2^n - 1), n >= 0. - Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Oct 18 2007, Jan 08 2008
2^(n+1) = 2*A005578(n) + 2*a(n) + 2*A000975(n-1). Let A005578(n), a(n), A000975(n-1) = triangle (a, b, c). Then ((S-c), (S-b), (S-a)) = (A005578(n-1), a(n-1), A000975(n-2)). Example: (a, b, c) = (11, 11, 10) = (A005578(5), a(5), A000975(4)). Then ((S-c), (S-b), (S-a)) = (6, 5, 5) = (A005578(4), a(4), A000975(3)). - Gary W. Adamson, Dec 24 2007
Sequence is identical to the absolute values of its inverse binomial transform. A similar result holds for [0,A001045*2^n]. - Paul Curtz, Jan 17 2008
From a(2) on (i.e., 1, 3, 5, 11, 21, ...) also: least odd number such that the subsets of {a(2), ..., a(n)} sum to 2^(n-1) different values, cf. A138000 and A064934. It is interesting to note the pattern of numbers occurring (or not occurring) as such a sum (A003158). - M. F. Hasler, Apr 09 2008
a(n) is the term (5, 1) of n-th power of the 5 X 5 matrix shown in A121231. - Gary W. Adamson, Oct 03 2008
A147612(a(n)) = 1. - Reinhard Zumkeller, Nov 08 2008
a(n+1) = Sum(A153778(i): 2^n <= i < 2^(n+1)). - Reinhard Zumkeller, Jan 01 2009
It appears that a(n) is also the number of integers between 2^n and 2^(n+1) that are divisible by 3 with no remainder. - John Fossaceca (john(AT)fossaceca.net), Jan 31 2009
Number of pairs of consecutive odious (or evil) numbers between 2^(n+1) and 2^(n+2), inclusive. - T. D. Noe, Feb 05 2009
Equals eigensequence of triangle A156319. - Gary W. Adamson, Feb 07 2009
A three-dimensional interpretation of a(n+1) is that it gives the number of ways of filling a 2 X 2 X n hole with 1 X 2 X 2 bricks. - Martin Griffiths, Mar 28 2009
Starting with offset 1 = INVERTi transform of A002605: (1, 2, 6, 16, 44, ...). - Gary W. Adamson, May 12 2009
Convolved with (1, 2, 2, 2, ...) = A000225: (1, 3, 7, 15, 31, ...). - Gary W. Adamson, May 23 2009
The product of a pair of successive terms is always a triangular number. - Giuseppe Ottonello, Jun 14 2009
Let A be the Hessenberg matrix of order n, defined by: A[1, j] = 1, A[i, i] := -2, A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^(n-1)*det(A). - Milan Janjic, Jan 26 2010
Let R denote the irreducible representation of the symmetric group S_3 of dimension 2, and let s and t denote respectively the sign and trivial irreducible representations of dimension 1. The decomposition of R^n into irreducible representations consists of a(n) copies of R and a(n-1) copies of each of s and t. - Andrew Rupinski, Mar 12 2010
As a fraction: 1/88 = 0.0113636363... or 1/9898 = 0.00010103051121... - Mark Dols, May 18 2010
Starting with "1" = the INVERT transform of (1, 0, 2, 0, 4, 0, 8, ...); e.g., a(7) = 43 = (1, 1, 1, 3, 5, 11, 21) dot (8, 0, 4, 0, 2, 0, 1) = (8 + 4 + 10 + 21) = 43. - Gary W. Adamson, Oct 28 2010
Rule 28 elementary cellular automaton (A266508) generates this sequence. - Paul Muljadi, Jan 27 2011
This is a divisibility sequence. - Michael Somos, Feb 06 2011
From L. Edson Jeffery, Apr 04 2011: (Start)
Let U be the unit-primitive matrix (see [Jeffery])
U = U_(6,2) =
  (0 0 1)
  (0 2 0)
  (2 0 1).
Then a(n+1) = (Trace(U^n))/3, a(n+1) = ((U^n){3, 3})/3, a(n) = ((U^n){1, 3})/3 and a(n) = ((U^(n+1))_{1, 1})/2. (End)
The sequence emerges in using iterated deletion of strictly dominated strategies to establish the best-response solution to the Cournot duopoly problem as a strictly dominant strategy. The best response of firm 1 to firm 2's chosen quantity is given by q*1 = 1/2*(a - c - q_2), where a is reservation price, c is marginal cost, and q_2 is firm two's chosen quantity. Given that q_2 is in [o, a - c], q*_1 must be in [o, 1/2*(a - c)]. Since costs are symmetric, we know q_2 is in [0, 1/2*(a - c)]. Then we know q*_1 is in [1/4*(a - c), 1/2*(a - c)]. Continuing in this way, the sequence of bounds we get (factoring out a - c) is {1/2, 1/4, 3/8, 5/16, ...}; the numerators are the Jacobsthal numbers. - _Michael Chirico, Sep 10 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 2, 3*a(n-1) equals the number of 3-colored compositions of n with all parts greater than or equal to 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011
This sequence is connected with the Collatz problem. We consider the array T(i, j) where the i-th row gives the parity trajectory of i, for example for i = 6, the infinite trajectory is 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 -> 4 -> 2 -> 1 -> 4 -> 2 -> 1... and T(6, j) = [0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, ..., 1, 0, 0, 1, ...]. Now, we consider the sum of the digits "1" of each column. We obtain the sequence a(n) = Sum_{k = 1..2^n} T(k, n) = Sum {k = 1..2^n} digits "1" of the n-th column. Because a(n) + a(n+1) = 2^n, then a(n+1) = Number of digits "0" among the 2^n elements of the n-th column. - _Michel Lagneau, Jan 11 2012
3!*a(n-1) is apparently the trace of the n-th power of the adjacency matrix of the complete 3-graph, a 3 X 3 matrix with diagonal elements all zero and off-diagonal all ones. The off-diagonal elements for the n-th power are all equal to a(n) while each diagonal element seems to be a(n) + 1 for an even power and a(n) - 1 for an odd. These are related to the lengths of closed paths on the graph (see Delfino and Viti's paper). - Tom Copeland, Nov 06 2012
From Paul Curtz, Dec 11 2012: (Start)
2^n * a(-n) = (-1)^(n-1) * a(n), which extends the sequence to negative indices: ..., -5/16, 3/8, -1/4, 1/2, 0, 1, 1, 3, 5, ...
The "autosequence" property with respect to the binomial transform mentioned in my comment of Jan 17 2008 is still valid if the term a(-1) is added to the array of the sequence and its iterated higher-order differences in subsequent rows:
    0   1/2  1/2  3/2  5/2 11/2 ...
   1/2   0    1    1    3    5 ...
  -1/2   1    0    2    2    6 ...
   3/2  -1    2    0    4    4 ...
  -5/2   3   -2    4    0    8 ...
  11/2  -5    6   -4    8    0 ...
The main diagonal in this array contains 0's. (End)
Assign to a triangle T(n, 0) = 1 and T(n+1, 1) = n; T(r, c) = T(r-1, c-1) + T(r-1, c-2) + T(r-2, c-2). Then T(n+1, n) - T(n, n) = a(n). - J. M. Bergot, May 02 2013
a(n+1) counts clockwise walks on n points on a circle that take steps of length 1 and 2, return to the starting point after two full circuits, and do not duplicate any steps (USAMO 2013, problem 5). - Kiran S. Kedlaya, May 11 2013
Define an infinite square array m by m(n, 0) = m(0, n) = a(n) in top row and left column and m(i, j) = m(i, j-1) + m(i-1, j-1) otherwise, then m(n+1, n+1) = 3^(n-1). - J. M. Bergot, May 10 2013
a(n) is the number of compositions (ordered partitions) of n - 1 into one sort of 1's and two sorts of 2's. Example: the a(4) = 5 compositions of 3 are 1 + 1 + 1, 1 + 2, 1 + 2', 2 + 1 and 2' + 1. - Bob Selcoe, Jun 24 2013
Without 0, a(n)/2^n equals the probability that n will occur as a partial sum in a randomly-generated infinite sequence of 1's and 2's. The limiting ratio is 2/3. - Bob Selcoe, Jul 04 2013
Number of conjugacy classes of Z/2Z X Z/2Z in GL(2,2^(n+1)). - Jared Warner, Aug 18 2013
a(n) is the top left entry of the (n-1)-st power of the 3 X 3 matrix [1, 1, 1, 1, 0, 0, 1, 0, 0]. a(n) is the top left entry of the (n+1)-st power of any of the six 3 X 3 matrices [0, 1, 0; 1, 1, 1; 0, 1, 0], [0, 1, 1; 0, 1, 1; 1, 1, 0], [0, 0, 1; 1, 1, 1; 1, 1, 0], [0, 1, 1; 1, 0, 1; 0, 1, 1], [0, 0, 1; 0, 0, 1; 1, 1, 1] or [0, 1, 0; 1, 0, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
This is the only integer sequence from the family of homogeneous linear recurrence of order 2 given by a(n) = k*a(n-1) + t*a(n-2) with positive integer coefficients k and t and initial values a(0) = 0 and a(1) = 1 whose ratio a(n+1)/a(n) converges to 2 as n approaches infinity. - Felix P. Muga II, Mar 14 2014
This is the Lucas sequence U(1, -2). - Felix P. Muga II, Mar 21 2014
sqrt(a(n+1) * a(n-1)) -> a(n) + 3/4 if n is even, and  -> a(n) - 3/4 if n is odd, for n >= 2. - Richard R. Forberg, Jun 24 2014
a(n+1) counts closed walks on the end vertices of P_3 containing one loop at the middle vertex. a(n-1) counts closed walks on the middle vertex of P_3 containing one loop at that vertex. - David Neil McGrath, Nov 07 2014
From César Eliud Lozada, Jan 21 2015: (Start)
Let P be a point in the plane of a triangle ABC (with sides a, b, c) and barycentric coordinates P = [x:y:z]. The Complement of P with respect to ABC is defined to be Complement(P) = [b*y + c*z : c*z + a*x : a*x + b*y].
Then, for n >= 1, Complement(Complement(...(Complement(P))..)) = (n times) =
      [2*a(n-1)*a*x + (2*a(n-1) - (-1)^n)*(b*y + c*z):
       2*a(n-1)*b*y + (2*a(n-1) - (-1)^n)*(c*z + a*x):
       2*a(n-1)*c*z + (2*a(n-1) - (-1)^n)*(a*x + b*y)]. (End)
a(n) (n >= 2) is the number of induced hypercubes of the Fibonacci cube Gamma(n-2). See p. 513 of the Klavzar reference. Example: a(5) = 11. Indeed, the Fibonacci cube Gamma(3) is <>- (cycle C(4) with a pendant edge) and the hypercubes are: 5 vertices, 5 edges, and 1 square. - Emeric Deutsch, Apr 07 2016
If the sequence of points {P_i(x_i, y_i)} on the cubic y = a*x^3 + b*x^2 + c*x + d has the property that the segment P_i(x_i, y_i) P_i+1(x_i+1, y_i+1) is always tangent to the cubic at P_i+1(x_i+1, y_i+1) then a(n) = -2^n*a/b*(x_(n+1)-(-1/2)^n*x_1). - Michael Brozinsky, Aug 01 2016
With the quantum integers defined by [n+1]A000225%20are%20given%20by%20q%20=%20sqrt(2).%20Cf.%20A239473.%20-%20_Tom%20Copeland">q = (q^(n+1) - q^(-n-1)) / (q - q^(-1)), the Jacobsthal numbers are a(n+1) = (-1)^n*q^n [n+1]_q with q = i * sqrt(2) for i^2 = -1, whereas the signed Mersenne numbers A000225 are given by q = sqrt(2). Cf. A239473. - _Tom Copeland, Sep 05 2016
Every positive integer has a unique expression as a sum of Jacobsthal numbers in which the index of the smallest summand is odd, with a(1) and a(2) both allowed. See the L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr. reference. - Ira M. Gessel, Dec 31 2016. See A280049 for these expansions. - N. J. A. Sloane, Dec 31 2016
For n > 0, a(n) equals the number of ternary words of length n-1 in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017
For n > 0, a(n) equals the number of orbits of the finite group PSL(2,2^n) acting on subsets of size 4 for the 2^n+1 points of the projective line. - Paul M. Bradley, Jan 31 2017
For n > 1, number of words of length n-2 over alphabet {1,2,3} such that no odd letter is followed by an odd letter. - Armend Shabani, Feb 17 2017
Also, the decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 678", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283641. - Robert Price, Mar 12 2017
Also the number of independent vertex sets and vertex covers in the 2 X (n-2) king graph. - Eric W. Weisstein, Sep 21 2017
From César Eliud Lozada, Dec 14 2017: (Start)
Let T(0) be a triangle and let T(1) be the medial triangle of T(0), T(2) the medial triangle of T(1) and, in general, T(n) the medial triangle of T(n-1). The barycentric coordinates of the first vertex of T(n) are [2*a(n-1)/a(n), 1, 1], for n > 0.
Let S(0) be a triangle and let S(1) be the antimedial triangle of S(0), S(2) the antimedial triangle of S(1) and, in general, S(n) the antimedial triangle of S(n-1). The barycentric coordinates of the first vertex of S(n) are [-a(n+1)/a(n), 1, 1], for n > 0. (End)
a(n) is also the number of derangements in S_{n+1} with empty peak set. - Isabella Huang, Apr 01 2018
For n > 0, gcd(a(n), a(n+1)) = 1. - Kengbo Lu, Jul 27 2020
Number of 2-compositions of n+1 with 1 not allowed as a part; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 17 2020
The number of Hamiltonian paths of the flower snark graph of even order 2n > 2 is 12*a(n-1). - Don Knuth, Dec 25 2020
When set S = {1, 2, ..., 2^n}, n>=0, then the largest subset T of S with the property that if x is in T, then 2*x is not in T, has a(n+1) elements. For example, for n = 4, #S = 16, a(5) = 11 with T = {1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16} (see Hassan Tarfaoui link, Concours Général 1991). - Bernard Schott, Feb 14 2022
a(n) is the number of words of length n over a binary alphabet whose position in the lexicographic order is one more than a multiple of three. a(3) = 3: aaa, abb, bba. - Alois P. Heinz, Apr 13 2022
Named by Horadam (1988) after the German mathematician Ernst Jacobsthal (1882-1965). - Amiram Eldar, Oct 02 2023
Define the sequence u(n) = (u(n-1) + u(n-2))/u(n-3) with u(0) = 0, u(1) = 1, u(2) = u(3) = -1. Then u(4*n) = -1 + (-1)^n/a(n+1), u(4*n+1) = 2 - (-1)^n/a(n+1), u(4*n+2) = u(4*n+3) = -1. For example, a(3) = 3 and u(8) = -2/3, u(9) = 5/3, u(10) = u(11) = -1. - Michael Somos, Oct 24 2023
From Miquel A. Fiol, May 25 2024: (Start)
Also, a(n) is the number of (3-color) states of a cycle (n+1)-pole C_{n+1} with n+1 terminals (or semiedges).
For instance, for n=3, the a(3)=3 states (3-coloring of the terminals) of C_4 are
  a a   a a   a b
  a a   b b   a b  (End)
Also, with offset 1, the cogrowth sequence of the 6-element dihedral group D3. - Sean A. Irvine, Nov 04 2024

			a(2) = 3 because the tiling of the 3 X 2 rectangle has either only 1 X 1 tiles, or one 2 X 2 tile in one of two positions (together with two 1 X 1 tiles).
From _Joerg Arndt_, Nov 10 2012: (Start)
The a(6)=21 length-5 ternary words with no two consecutive letters nonzero are (dots for 0's)
[ 1]   [ . . . . ]
[ 2]   [ . . . 1 ]
[ 3]   [ . . . 2 ]
[ 4]   [ . . 1 . ]
[ 5]   [ . . 2 . ]
[ 6]   [ . 1 . . ]
[ 7]   [ . 1 . 1 ]
[ 8]   [ . 1 . 2 ]
[ 9]   [ . 2 . . ]
[10]   [ . 2 . 1 ]
[11]   [ . 2 . 2 ]
[12]   [ 1 . . . ]
[13]   [ 1 . . 1 ]
[14]   [ 1 . . 2 ]
[15]   [ 1 . 1 . ]
[16]   [ 1 . 2 . ]
[17]   [ 2 . . . ]
[18]   [ 2 . . 1 ]
[19]   [ 2 . . 2 ]
[20]   [ 2 . 1 . ]
[21]   [ 2 . 2 . ]
(End)
G.f. = x + x^2 + 3*x^3 + 5*x^4 + 11*x^5 + 21*x^6 + 43*x^7 + 85*x^8 + 171*x^9 + ...

Jathan Austin and Lisa Schneider, Generalized Fibonacci sequences in Pythagorean triple preserving sequences, Fib. Q., 58:1 (2020), 340-350.
Thomas Fink and Yong Mao, The 85 ways to tie a tie, Fourth Estate, London, 1999; Die 85 Methoden eine Krawatte zu binden. Hoffmann und Kampe, Hamburg, 1999.
International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest Problem #16.
Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World Scientific Press, 2007. See p. 80.
Ernst Erich Jacobsthal, Fibonaccische Polynome und Kreisteilungsgleichungen, Sitzungsber. Berliner Math. Gesell. 17 (1919-1920), 43-57.
Tanya Khovanova, "Coins and Logic", Chapter 6, The Mathematics of Various Entertaining Subjects: Volume 3 (2019), Jennifer Beineke & Jason Rosenhouse, eds. Princeton University Press, Princeton and Oxford, p. 73. ISBN: 0691182582, 978-0691182582.
Donald E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.3.1, Eq. 13.
Thomas Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 98.
Steven Roman, Introduction to Coding and Information Theory, Springer Verlag, 1996, 41-42.
P. D. Seymour and D. J. A. Welsh, Combinatorial applications of an inequality form statistical mechanics, Math. Proc. Camb. Phil. Soc. 77 (1975), 485-495. [Although Daykin et al. (1979) claim that the present sequence is studied in this article, it does not seem to be explicitly mentioned. Note that definition of log-convex in (3.1) is wrong. - N. J. A. Sloane, Dec 26 2020]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Robert M. Young, Excursions in Calculus, MAA, 1992, p. 239

Indranil Ghosh, Table of n, a(n) for n = 0..3316 (terms 0..500 from T. D. Noe)
M. H. Albert, M. D. Atkinson, and V. Vatter, Inflations of geometric grid classes: three case studies.
Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, and Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
Joerg Arndt, Matters Computational (The Fxtbook), pp.315-318.
Mohammad K. Azarian, Limit of Nested Square Roots, Problem B-664, Fibonacci Quarterly, Vol. 28, No. 2, May 1990, p. 182.
Mohammad K. Azarian, Solution to Problem B-664, Limit of Nested Square Roots, Fibonacci Quarterly, Vol. 29, No. 2, May 1991, p. 182.
Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015.
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19 (2016), Article 16.3.5.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 14, 29.
Katerina Böhmová, Cristina Dalfó, and Clemens Huemer, On cyclic Kautz digraphs, arXiv:1512.05917 [math.CO], 2015; alternative link.
Wieb Bosma, Signed bits and fast exponentiation, J. Th. Nombres de Bordeaux, 13 no. 1 (2001), p. 27-41.
Garry Bowlin and Matthew G. Brin, Coloring Planar Graphs via Colored Paths in the Associahedra, arXiv preprint arXiv:1301.3984 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 12 2013
Rachael Boyd and Richard Hepworth, Combinatorics of injective words for Temperley-Lieb algebras, arXiv:2006.04261 [math.AT], 2020.
Paul Bradley and Peter Rowley, Orbits on k-subsets of 2-transitive Simple Lie-type Groups, 2014.
Henri Brocard, Propriété d'une série de triangles, Nouvelle Correspondance Mathématique, Vol. 6 (1880), 145-151.
H. Bruhn, L. Gellert, and J. Günther, Jacobsthal numbers in generalised Petersen graphs, arXiv preprint arXiv:1503.03390 [math.CO], 2015.
H. Bruhn, L. Gellert, and J. Günther, Jacobsthal numbers in generalised Petersen graphs, Electronic Notes in Discrete Math., 2015.
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Intrinsic Properties of a Non-Symmetric Number Triangle, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Representations for a special sequence, Fibonacci Quarterly 10.5 (1972), pages 499-518 and 550.
Paula Catarino, Helena Campos, and Paulo Vasco. On the Mersenne sequence. Annales Mathematicae et Informaticae, 46 (2016) pp. 37-53.
Miquel Cerda, Irregular triangle whose row sums are the Jacobsthal numbers.
Ji Young Choi, Ternary Modified Collatz Sequences And Jacobsthal Numbers, Journal of Integer Sequences, Vol. 19 (2016), #16.7.5.
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
M. H. Cilasun, An Analytical Approach to Exponent-Restricted Multiple Counting Sequences, arXiv preprint arXiv:1412.3265 [math.NT], 2014.
M. H. Cilasun, Generalized Multiple Counting Jacobsthal Sequences of Fermat Pseudoprimes, Journal of Integer Sequences, Vol. 19, 2016, #16.2.3.
Charles K. Cook and Michael R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41 (2013) pp. 27-39.
D. E. Daykin, D. J. Kleitman, and D. B. West, The number of meets between two subsets of a lattice, J. Combin. Theory, A 26 (1979), 135-156.
Gesualdo Delfino and Jacopo Viti, Potts q-color field theory and scaling random cluster model, arXiv:1104.4323 [hep-th], 2011.
Karl Dilcher and Larry Ericksen, Continued fractions and Stern polynomials. Ramanujan Journal (2017). See Table 2.
Karl Dilcher and Hayley Tomkins, Square classes and divisibility properties of Stern polynomials, Integers (2018) 18, Article #A29.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
Peter E. Finch, From spin to anyon notation: The XXZ Heisenberg model as a D_3 (or su(2)_4) anyon chain, arXiv preprint arXiv:1201.4470 [math-ph], 2012.
M. A. Fiol and J. Vilaltella, Some results on the structure of multipoles in the study of snarks, Electron J. Combin., 22(1) (2015), #P1.45.
M. Cetin Firengiz and A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
Darrin D. Frey and James A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, J. Integer Seqs., Vol. 3 (2000), Article 00.2.3.
Robert Frontczak and Taras Goy, Mersenne-Horadam identities using generating functions, Carpathian Math. Publ. (2020) Vol. 12, No. 1, 34-45.
E. M. García-Caballero, S. G. Moreno, and M. P. Prophet, New Viète-like infinite products of nested radicals, Fib. Q., 52 (No. 1, 2014), 27-31.
Dale Gerdemann, Triangles from (1,2) Recursion and Recursive Triangle Fragmentation from (1,2) Recursion , YouTube Videos, December 2014.
A. Goldberg and N. A. Rosenberg, Beyond 2/3 and 1/3: the complex signatures of sex-biased admixture on the X chromosome, Genetics 201 (2015), 263-279.
J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete Math., 204 (1999), 231-236.
Taras Goy, On determinants and permanents of some Toeplitz-Hessenberg matrices whose entries are Jacobsthal numbers, Eurasian Math. J. (2018) Vol. 9, No. 4, 61-67.
Taras P. Goy, Jacobsthal number identities using the generalized Brioschi formula, Vasyl Stefanyk Precarpathian National University, (Ivano-Frankivsk, Ukraine, 2020).
Jonny Griffiths, The Not-on-the-line transformation, The Mathematical Gazette, 99, pp 347-352 (2015).
Martin Griffiths and Alex Bramham, The Jacobsthal Numbers: Two Results and Two Questions, Fibonacci Quart. 53 (2015), no. 2, 147-151.
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968.
Richard K. Guy, Tanya Khovanova, and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099 [math.NT], 2012-2014.
Lee Hae-hwang, Illustration of initial terms in terms of rosemary plants.
Silvia Heubach, Tiling an m-by-n area with squares of size up to k-by-k (m<=5), preprint, published in: Congressus Numerantium 140 (1999), 43-64.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 56. Book's website.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
Brian Hopkins, Mark Shattuck, Andrew V. Sills, Thotsaporn "AEK" Thanatipanonda, and Hua Wang, Parts and subword patterns in compositions, Preprint 2015.
A. F. Horadam, Jacobsthal and Pell Curves, Fib. Quart. 26, 79-83, 1988. [See J_n.]
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart. 34, 40-54, 1996.
Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 17.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 142.
Dan Ismailescu, Joehyun Kim, Kelvin Kim, and Jeewoo Lee, The largest angle bisection procedure, arXiv:1908.02749 [math.MG], 2019. See p. 17.
Milan Janjić, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015), Article 15.4.7.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), Article 18.1.4.
L. Edson Jeffery, Unit-primitive matrices.
D. Jhala, G. P. S. Rathore, and K. Sisodiya, Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124.
Tanya Khovanova, Coins and Logic, arXiv:1801.01143 [math.HO], 2018.
Tanya Khovanova and Konstantin Knop, Coins that Change Their Weights, arXiv:1611.09201 [math.CO], 2016.
Sandi Klavžar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
Masato Kobayashi, Weighted counting of Bruhat paths by shifted R-polynomials, arXiv:1907.11802 [math.CO], 2019.
Takao Komatsu, Continued fractions associated with the topological index of the caterpillar-bond graph, arXiv:1903.09986 [math.CO], 2019.
Thomas Koshy and Ralph P. Grimaldi, Ternary words and Jacobsthal numbers, Fib. Quart., 55 (No. 2, 2017), 129-136.
Peter J. Larcombe, Julius Fergy T. Rabago, and Eric J. Fennessey, On two derivative sequences from scaled geometric mean sequence terms, Palestine Journal of Mathematics (2018) Vol. 7(2), 397-405.
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
E. Lemmen, J. van Duivenbode, J. L. Duarte, and E. A. Lomonova, Flexible Multilevel Converters using 4-Switch Extended Commutation Cells, Emerging and Selected Topics in Power Electronics, IEEE Journal of, Volume:PP, Issue: 99, 2015.
S. L. Levine, Suppose more rabbits are born, Fib. Quart., 26 (1988), 306-311.
Alan J. Macfarlane, On generating functions of some sequences of integers defined in the evolution of the cellular automaton Rule 150, Preprint 2016.
T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, arXiv:math/0204005 [math.CO], 2002.
B. D. McKay and I. M. Wanless, A census of small latin hypercubes, SIAM J. Discrete Math. 22, (2008) 719-736.
A. Moghaddamfar and H. Tajbakhsh, More Determinant Representations for Sequences, Journal of Integer Sequences, 17 (2014), #14.5.6.
Sophie Morier-Genoud, Counting Coxeter's friezes over a finite field via moduli spaces, arXiv:1907.12790 [math.CO], 2019.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Emanuele Munarini, A generalization of André-Jeannin's symmetric identity, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 98-118.
G. Myerson and A. J. van der Poorten, Some problems concerning recurrence sequences, Amer. Math. Monthly 102 (1995), no. 8, 698-705.
S. Northshield, Stern's diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ..., Amer. Math. Monthly, 117 (2010), 581-598.
Kritkhajohn Onphaeng and Prapanpong Pongsriiam, Jacobsthal and Jacobsthal-Lucas Numbers and Sums Introduced by Jacobsthal and Tverberg, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.6.
Ahmet Öteleş, Zekeriya Y. Karatas, and Diyar O. Mustafa Zangana, Jacobsthal Numbers and Associated Hessenberg Matrices, J. Int. Seq., Vol. 21 (2018), Article 18.2.5.
D. Panario, M. Sahin, and Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.
D. Panario, M. Sahin, Q. Wang, and W. Webb, General conditional recurrences, Applied Mathematics and Computation, Volume 243, Sep 15 2014, Pages 220-231.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
M. Rahmani, The Akiyama-Tanigawa matrix and related combinatorial identities, Linear Algebra and its Applications 438 (2013) 219-230. - From _N. J. A. Sloane_, Dec 26 2012
N. A. Rosenberg, Admixture models and the breeding systems of H. S. Jennings: a GENETICS connection, Genetics 202 (2016), 9-13.
E. Schlemm, On the expected number of successes in a sequence of nested Bernoulli trials, arXiv preprint arXiv:1303.4979 [math.PR], 2013.
A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, Vol. 21, 2015, No. 2, 35-42.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Yüksel Soykan, On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research (2019) Vol. 20, No. 2, 1-15, Article AIR.51824.
Yüksel Soykan, Generalized Fibonacci Numbers: Sum Formulas, Journal of Advances in Mathematics and Computer Science (2020) Vol. 35, No. 1, 89-104.
Yüksel Soykan, Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers, Asian Journal of Advanced Research and Reports (2020) Vol. 9, No. 1, 23-39, Article no. AJARR.55441.
Yüksel Soykan, A Study on Generalized Fibonacci Numbers: Sum Formulas Sum_{k=0..n} k * x^k * W_k^3 and Sum_{k=1..n} k * x^k W_-k^3 for the Cubes of Terms, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 297-331.
Yüksel Soykan, Erkan Taşdemir, and İnci Okumuş, On Dual Hyperbolic Numbers With Generalized Jacobsthal Numbers Components, Zonguldak Bülent Ecevit University, (Zonguldak, Turkey, 2019).
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv:1504.04404 [math.CO], 2015.
X. Sun and V. H. Moll, The p-adic Valuations of Sequences Counting Alternating Sign Matrices, JIS 12 (2009) 09.3.8.
Hassan Tarfaoui, Concours Général 1991 - Exercice 4 (in French).
Two-Year College Math. Jnl., Review of "Jacobsthal Representation Numbers", 28 (1997), p. 76.
USA Mathematical Olympiad 2013, Problem 2 (proposed by Sam Vandervelde).
Eric Weisstein's World of Mathematics, Jacobsthal Number, Negabinary, Repunit, Rule 28.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, King Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
Wikipedia, Elementary cellular automaton
OEIS Wiki, Autosequence.
Abdelmoumène Zekiri, Farid Bencherif, and Rachid Boumahdi, Generalization of an Identity of Apostol, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.
G. B. M. Zerr, Problem 64, American Mathematical Monthly, vol. 3, no. 12, 1896 (p. 311).
Index entries for "core" sequences.
Index entries for linear recurrences with constant coefficients, signature (1,2).
Index entries for sequences related to Chebyshev polynomials.
Index to divisibility sequences.
Index to sequences related to Olympiads and other Mathematical competitions.

Partial sums of this sequence give A000975, where there are additional comments from B. E. Williams and Bill Blewett on the tie problem.
A002487(a(n)) = A000045(n).
Row sums of A059260, A156667 and A134317. Equals A026644(n-2)+1 for n > 1.
a(n) = A073370(n-1, 0), n >= 1 (first column of triangle).
Cf. A266508 (binary), A081857 (base 4), A147612 (characteristic function).
Cf. A000978, A000979, A019322, A066845, A105348, A130249, A130250, A130253, A005578, A002083, A113405, A138000, A064934, A003158, A175286 (Pisano periods), A147613, A156319, A002605, A000225, A052129, A014551 (companion "autosequence"), A015266, A015287, A015305, A015323, A015338, A015356, A015371, A084175, A245489, A283641.
Cf. A049883 = primes in this sequence, A107036 = indices of primes, A129738.
Cf. A091084 (mod 10), A239473, A280049.
Bisections: A002450, A007583.
Cf. A077925 (signed version).

a001045 = (`div` 3) . (+ 1) . a000079
a001045_list = 0 : 1 :
   zipWith (+) (map (2 *) a001045_list) (tail a001045_list)
-- Reinhard Zumkeller, Mar 24 2013, Jan 05 2012, Feb 05 2011

[n le 2 select n-1 else Self(n-1)+2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jun 27 2016

A001045 := proc(n)
  (2^n-(-1)^n)/3 ;
end proc: # R. J. Mathar, Dec 18 2012

Jacob0[n_] := (2^n - (-1)^n)/3; Table[Jacob0[n], {n, 0, 33}] (* Robert G. Wilson v, Dec 05 2005 *)
Array[(2^# - (-1)^#)/3 &, 33, 0] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
LinearRecurrence[{1, 2}, {0, 1}, 40] (* Harvey P. Dale, Nov 30 2011 *)
CoefficientList[Series[x/(1 - x - 2 x^2), {x, 0, 34}], x] (* Robert G. Wilson v, Jul 21 2015 *)
Table[(2^n - (-1)^n)/3, {n, 0, 20}] (* Eric W. Weisstein, Sep 21 2017 *)
Table[Abs[QBinomial[n, 1, -2]], {n, 0, 35}] (* John Keith, Jan 29 2022 *)

a[0]:0$
a[n]:=2^(n-1)-a[n-1]$
A001045(n):=a[n]$
makelist(A001045(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
a(n) = (2^n - (-1)^n) / 3
```

M=[1,1,0;1,0,1;0,1,1];for(i=0,34,print1((M^i)[2,1],",")) \\ Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 28 2005

a=0; for(n=0,34,print1(a,", "); a=2*(a-n%2)+1) \\ K. Spage, Aug 22 2014

# A001045.py
def A001045():
    a, b = 0, 1
    while True:
        yield a
        a, b = b, b+2*a
sequence = A001045()
[next(sequence) for i in range(20)] # David Radcliffe, Jun 26 2016

[(2**n-(-1)**n)//3 for n in range(40)] # Gennady Eremin, Mar 03 2022

[lucas_number1(n, 1, -2) for n in range(34)]  # Zerinvary Lajos, Apr 22 2009
# Alternatively:
a = BinaryRecurrenceSequence(1,2)
[a(n) for n in (0..34)] # Peter Luschny, Aug 29 2016

a(n) = 2^(n-1) - a(n-1). a(n) = 2*a(n-1) - (-1)^n = (2^n - (-1)^n)/3.
G.f.: x/(1 - x - 2*x^2) = x/((x+1)*(1-2*x)). Simon Plouffe in his 1992 dissertation
E.g.f.: (exp(2*x) - exp(-x))/3.
a(2*n) = 2*a(2*n-1)-1 for n >= 1, a(2*n+1) = 2*a(2*n)+1 for n >= 0. - Lee Hae-hwang, Oct 11 2002; corrected by Mario Catalani (mario.catalani(AT)unito.it), Dec 04 2002
Also a(n) is the coefficient of x^(n-1) in the bivariate Fibonacci polynomials F(n)(x, y) = x*F(n-1)(x, y) + y*F(n-2)(x, y), with y=2*x^2. - Mario Catalani (mario.catalani(AT)unito.it), Dec 04 2002
a(n) = Sum_{k=1..n} binomial(n, k)*(-1)^(n+k)*3^(k-1). - Paul Barry, Apr 02 2003
The ratios a(n)/2^(n-1) converge to 2/3 and every fraction after 1/2 is the arithmetic mean of the two preceding fractions. - Gary W. Adamson, Jul 05 2003
a(n) = U(n-1, i/(2*sqrt(2)))*(-i*sqrt(2))^(n-1) with i^2=-1. - Paul Barry, Nov 17 2003
a(n+1) = Sum_{k=0..ceiling(n/2)} 2^k*binomial(n-k, k). - Benoit Cloitre, Mar 06 2004
a(2*n) = A002450(n) = (4^n - 1)/3; a(2*n+1) = A007583(n) = (2^(2*n+1) + 1)/3. - Philippe Deléham, Mar 27 2004
a(n) = round(2^n/3) = (2^n + (-1)^(n-1))/3 so lim_{n->infinity} 2^n/a(n) = 3. - Gerald McGarvey, Jul 21 2004
a(n) = Sum_{k=0..n-1} (-1)^k*2^(n-k-1) = Sum_{k=0..n-1}, 2^k*(-1)^(n-k-1). - Paul Barry, Jul 30 2004
a(n+1) = Sum_{k=0..n} binomial(k, n-k)*2^(n-k). - Paul Barry, Oct 07 2004
a(n) = Sum_{k=0..n-1} W(n-k, k)*(-1)^(n-k)*binomial(2*k,k), W(n, k) as in A004070. - Paul Barry, Dec 17 2004
From Paul Barry, Jan 17 2005: (Start)
a(n) = Sum_{k=0..n} k*binomial(n-1, (n-k)/2)*(1+(-1)^(n+k))*floor((2*k+1)/3).
a(n+1) = Sum_{k=0..n} k*binomial(n-1, (n-k)/2)*(1+(-1)^(n+k))*(A042965(k)+0^k). (End)
From Paul Barry, Jan 17 2005: (Start)
a(n+1) = ceiling(2^n/3) + floor(2^n/3) = (ceiling(2^n/3))^2 - (floor(2^n/3))^2.
a(n+1) = A005578(n) + A000975(n-1) = A005578(n)^2 - A000975(n-1)^2. (End)
a(n+1) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*binomial(j, k). - Paul Barry, Jan 26 2005
Let M = [1, 1, 0; 1, 0, 1; 0, 1, 1], then a(n) = (M^n)[2, 1], also matrix characteristic polynomial x^3 - 2*x^2 - x + 2 defines the three-step recursion a(0)=0, a(1)=1, a(2)=1, a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n > 2. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 28 2005
a(n) = ceiling(2^(n+1)/3) - ceiling(2^n/3) = A005578(n+1) - A005578(n). - Paul Barry, Oct 08 2005
a(n) = floor(2^(n+1)/3) - floor(2^n/3) = A000975(n) - A000975(n-1). - Paul Barry, Oct 08 2005
From Paul Barry, Feb 20 2003: (Start)
a(n) = Sum_{k=0..floor(n/3)} binomial(n, f(n-1)+3*k);
a(n) = Sum_{k=0..floor(n/3)} binomial(n, f(n-2)+3*k), where f(n)=A080425(n). (End)
From Miklos Kristof, Mar 07 2007: (Start)
a(2*n) = (1/3)*Product_{d|n} cyclotomic(d,4).
a(2*n+1) = (1/3)*Product_{d|2*n+1} cyclotomic(2*d,2). (End)
From Hieronymus Fischer, Apr 23 2007: (Start)
The a(n) are closely related to nested square roots; this is 2*sin(2^(-n)*Pi/2*a(n)) = sqrt(2-sqrt(2-sqrt(2-sqrt(...sqrt(2)))...) {with the '2' n times, n >= 0}.
Also 2*cos(2^(-n)*Pi*a(n)) = sqrt(2-sqrt(2-sqrt(2-sqrt(...sqrt(2)))...) {with the '2' n-1 times, n >= 1} as well as
2*sin(2^(-n)*3/2*Pi*a(n)) = sqrt(2+sqrt(2+sqrt(2+sqrt(...sqrt(2)))...) {with the '2' n times, n >= 0} and
2*cos(2^(-n)*3*Pi*a(n)) = -sqrt(2+sqrt(2+sqrt(2+sqrt(...sqrt(2)))...) {with the '2' n-1 times, n >= 1}.
a(n) = 2^(n+1)/Pi*arcsin(b(n+1)/2) where b(n) is defined recursively by b(0)=2, b(n)=sqrt(2-b(n-1)).
There is a similar formula regarding the arccos function, this is a(n) = 2^n/Pi*arccos(b(n)/2).
With respect to the sequence c(n) defined recursively by c(0)=-2, c(n)=sqrt(2+c(n-1)), the following formulas hold true: a(n) = 2^n/3*(1-(-1)^n*(1-2/Pi*arcsin(c(n+1)/2))); a(n) = 2^n/3*(1-(-1)^n*(1-1/Pi*arccos(-c(n)/2))).
(End)
Sum_{k=0..n} A039599(n,k)*a(k) = A049027(n), for n >= 1. - Philippe Deléham, Jun 10 2007
Sum_{k=0..n} A039599(n,k)*a(k+1) = A067336(n). - Philippe Deléham, Jun 10 2007
Let T = the 3 X 3 matrix [1,1,0; 1,0,1; 0,1,1]. Then T^n * [1,0,0,] = [A005578(n), a(n), A000975(n-1)]. - Gary W. Adamson, Dec 24 2007
a(n) + a(n+5) = 11*2^n. - Paul Curtz, Jan 17 2008
a(n) = Sum_{k=1..n} K(2, k)*a(n - k), where K(n,k) = k if 0 <= k <= n and K(n,k)=0 otherwise. (When using such a K-coefficient, several different arguments to K or several different definitions of K may lead to the same integer sequence. For example, the Fibonacci sequence can be generated in several ways using the K-coefficient.) - Thomas Wieder, Jan 13 2008
a(n) + a(n+2*k+1) = a(2*k+1)*2^n. - Paul Curtz, Feb 12 2008
a(n) = lower left term in the 2 X 2 matrix [0,2; 1,1]^n. - Gary W. Adamson, Mar 02 2008
a(n+1) = Sum_{k=0..n} A109466(n,k)*(-2)^(n-k). -Philippe Deléham, Oct 26 2008
a(n) = sqrt(8*a(n-1)*a(n-2) + 1). E.g., sqrt(3*5*8+1) = 11, sqrt(5*11*8+1) = 21. - Giuseppe Ottonello, Jun 14 2009
Let p[i] = Fibonacci(i-1) and let A be the Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i = j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n-1) = det(A). - Milan Janjic, May 08 2010
a(p-1) = p*A007663(n)/3 if n > 1, and a(p-1) = p*A096060(n) if n > 2, with p=prime(n). - Jonathan Sondow, Jul 19 2010
Algebraically equivalent to replacing the 5's with 9's in the explicit (Binet) formula for the n-th term in the Fibonacci sequence: The formula for the n-th term in the Fibonacci sequence is F(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)). Replacing the 5's with 9's gives ((1+sqrt(9))^n - (1-sqrt(9))^n)/(2^n*sqrt(9)) = (2^n+(-1)^(n+1))/3 = (2^n-(-1)^(n))/3 = a(n). - Jeffrey R. Goodwin, May 27 2011
For n > 1, a(n) = A000975(n-1) + (1 + (-1)^(n-1))/2. - Vladimir Shevelev, Feb 27 2012
From Sergei N. Gladkovskii, Jun 12 2012: (Start)
G.f.: x/(1-x-2*x^2) = G(0)/3; G(k) = 1 - ((-1)^k)/(2^k - 2*x*4^k/(2*x*2^k - ((-1)^k)/G(k+1))); (continued fraction 3 kind, 3-step).
E.g.f.: G(0)/3; G(k) = 1 - ((-1)^k)/(2^k - 2*x*4^k/(2*x*2^k - ((-1)^k)*(k+1)/G(k+1))); (continued fraction 3rd kind, 3-step). (End)
a(n) = 2^k * a(n-k) + (-1)^(n+k)*a(k). - Paul Curtz, Jean-François Alcover, Dec 11 2012
a(n) = sqrt((A014551(n))^2 + (-1)^(n-1)*2^(n+2))/3. - Vladimir Shevelev, Mar 13 2013
G.f.: Q(0)/3, where Q(k) = 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(2*k+1 + 2*x)/( x*(2*k+2 + 2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 29 2013
G.f.: Q(0) -1, where Q(k) = 1 + 2*x^2 + (k+2)*x - x*(k+1 + 2*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 06 2013
a(n+2) = Sum_{k=0..n} A108561(n,k)*(-2)^k. - Philippe Deléham, Nov 17 2013
a(n) = (Sum_{k=1..n, k odd} C(n,k)*3^(k-1))/2^(n-1). - Vladimir Shevelev, Feb 05 2014
a(-n) = -(-1)^n * a(n) / 2^n for all n in Z. - Michael Somos, Mar 18 2014
a(n) = (-1)^(n-1)*Sum_{k=0..n-1} A135278(n-1,k)*(-3)^k = (2^n - (-1)^n)/3 = (-1)^(n-1)*Sum_{k=0..n-1} (-2)^k. Equals (-1)^(n-1)*Phi(n,-2), where Phi is the cyclotomic polynomial when n is an odd prime. (For n > 0.) - Tom Copeland, Apr 14 2014
From Peter Bala, Apr 06 2015: (Start)
a(2*n)/a(n) = A014551(n) for n >= 1; a(3*n)/a(n) = 3*A245489(n) for n >= 1.
exp( Sum_{n >= 1} a(2*n)/a(n)*x^n/n ) = Sum_{n >= 0} a(n+1)*x^n.
exp( Sum_{n >= 1} a(3*n)/a(n)*x^n/n ) = Sum_{n >= 0} A084175(n+1)*x^n.
exp( Sum_{n >= 1} a(4*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015266(n+3)*(-x)^n.
exp( Sum_{n >= 1} a(5*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015287(n+4)*x^n.
exp( Sum_{n >= 1} a(6*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015305(n+5)*(-x)^n.
exp( Sum_{n >= 1} a(7*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015323(n+6)*x^n.
exp( Sum_{n >= 1} a(8*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015338(n+7)*(-x)^n.
exp( Sum_{n >= 1} a(9*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015356(n+8)*x^n.
exp( Sum_{n >= 1} a(10*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015371(n+9)*(-x)^n. (End)
a(n) = (1-(-1)^n)/2 + floor((2^n)/3). - Reiner Moewald, Jun 05 2015
a(n+k)^2 - A014551(k)*a(n)*a(n+k) + (-2)^k*a(n)^2 = (-2)^n*a(k)^2, for n >= 0 and k >= 0. - Alexander Samokrutov, Jul 21 2015
Dirichlet g.f.: (PolyLog(s,2) + (1 - 2^(1-s))*zeta(s))/3. - Ilya Gutkovskiy, Jun 27 2016
From Yuchun Ji, Apr 08 2018: (Start)
a(m)*a(n) + a(m-1)*a(n-1) - 2*a(m-2)*a(n-2) = 2^(m+n-3).
a(m+n-1) = a(m)*a(n) + 2*a(m-1)*a(n-1); a(m+n) = a(m+1)*a(n+1) - 4*a(m-1)*a(n-1).
a(2*n-1) = a(n)^2 + 2*a(n-1)^2; a(2*n) = a(n+1)^2 - 4*a(n-1)^2. (End)
a(n+4) = a(n) + 5*2^n, a(0) = 0, a(1..4) = [1,1,3,5]. That is to say, for n > 0, the ones digits of Jacobsthal numbers follow the pattern 1,1,3,5,1,1,3,5,1,1,3,5,.... - Yuchun Ji, Apr 25 2019
a(n) mod 10 = A091084(n). - Alois P. Heinz, Apr 25 2019
The sequence starting with "1" is the second INVERT transform of (1, -1, 3, -5, 11, -21, 43, ...). - Gary W. Adamson, Jul 08 2019
From Kai Wang, Jan 14 2020: (Start)
a(n)^2 - a(n+1)*a(n-1) = (-2)^(n-1).
a(n)^2 - a(n+r)*a(n-r) = (-2)^(n-r)*a(r)^2.
a(m)*a(n+1) - a(m+1)*a(n) = (-2)^n*a(m-n).
a(m-n) = (-1)^n*(a(m)*A014551(n) - A014551(m)*a(n))/(2^(n+1)).
a(m+n) = (a(m)*A014551(n) + A014551(m)*a(n))/2.
A014551(n)^2 - A014551(n+r)*A014551(n-r) = 9*(-1)^(n-r-1)*2^(n-r)*a(r)^2 .
A014551(m)*A014551(n+1) - A014551(m+1)*A014551(n) = 9*(-1)^(n-1)*2^(n)*a(m-n).
A014551(m-n) = (-1)^(n)*(A014551(m)*A014551(n) - 9*a(m)*a(n))/2^(n+1).
A014551(m+n) = (A014551(m)*A014551(n) + 9*a(m)*a(n))/2.
a(n) = Sum_{i=0..n-1; j=0..n-1; i+2*j=n-1} 2^j*((i+j)!/(i!*j!)). (End)
For n > 0, 1/(2*a(n+1)) = Sum_{m>=n} a(m)/(a(m+1)*a(m+2)). - Kai Wang, Mar 03 2020
For 4 > h >= 0, k >= 0, a(4*k+h) mod 5 = a(h) mod 5. - Kai Wang, May 07 2020
From Kengbo Lu, Jul 27 2020: (Start)
a(n) = 1 + Sum_{k=0..n-1} a(k) if n odd; a(n) = Sum_{k=0..n-1} a(k) if n even.
a(n) = F(n) + Sum_{k=0..n-2} a(k)*F(n-k-1), where F denotes the Fibonacci numbers.
a(n) = b(n) + Sum_{k=0..n-1} a(k)*b(n-k), where b(n) is defined through b(0) = 0, b(1) = 1, b(n) = 2*b(n-2).
a(n) = 1 + 2*Sum_{k=0..n-2} a(k).
a(m+n) = a(m)*a(n+1) + 2*a(m-1)*a(n).
a(2*n) = Sum_{i>=0, j>=0} binomial(n-j-1,i)*binomial(n-i-1,j)*2^(i+j). (End)
G.f.: x/(1 - x - 2*x^2) = Sum_{n >= 0} x^(n+1) * Product_{k = 1..n} (k + 2*x)/(1 + k*x) (a telescoping series). - Peter Bala, May 08 2024
a(n) = Sum_{r>=0} F(n-2r, r), where F(n, 0) is the n-th Fibonacci number and F(n,r) = Sum_{j=1..n} F(n+1-j, r-1) F(j, r-1). - Gregory L. Simay, Aug 31 2024
From Peter Bala, Jun 27 2025: (Start)
The following are all examples of telescoping infinite products:
Product_{n >= 1} (1 + 2^n/a(2*n+2)) = 2, since 1 + 2^n/a(2*n+2) = b(n+1)/b(n), where b(n) = 2 - 3/(2^n + 1).
Product_{n >= 1} (1 - 2^n/a(2*n+2)) = 2/5, since 1 - 2^n/a(2*n+2) = c(n+1)/c(n), where c(n) = 2 + 3/(2^n - 1).
Product_{n >= 1} (1 + (-2)^n/a(2*n+2)) = 2/3, since 1 + (-2)^n/a(2*n+2) = d(n+1)/d(n), where d(n) = 2 - 1/(1 + (-2)^n).
Product_{n >= 1} (1 - (-2)^n/a(2*n+2)) = 6/5, since 1 - (-2)^n/a(2*n+2) = e(n+1)/e(n), where e(n) = 2 - 1/(1 - (-2)^n). (End)

Thanks to Don Knuth, who pointed out several missing references, including Brocard (1880), which although it was mentioned in the 1973 Handbook of Integer Sequences, was omitted from the 1995 "Encyclopedia". - N. J. A. Sloane, Dec 26 2020

A000071 a(n) = Fibonacci(n) - 1.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583, 4180, 6764, 10945, 17710, 28656, 46367, 75024, 121392, 196417, 317810, 514228, 832039, 1346268, 2178308, 3524577, 5702886, 9227464, 14930351, 24157816, 39088168, 63245985, 102334154

Offset: 1

N. J. A. Sloane

a(n) is the number of allowable transition rules for passing from one change to the next (on n-1 bells) in the English art of bell-ringing. This is also the number of involutions in the symmetric group S_{n-1} which can be represented as a product of transpositions of consecutive numbers from {1, 2, ..., n-1}. Thus for n = 6 we have a(6) from (12), (12)(34), (12)(45), (23), (23)(45), (34), (45), for instance. See my 1983 Math. Proc. Camb. Phil. Soc. paper. - Arthur T. White, letter to N. J. A. Sloane, Dec 18 1986
Number of permutations p of {1, 2, ..., n-1} such that max|p(i) - i| = 1. Example: a(4) = 2 since only the permutations 132 and 213 of {1, 2, 3} satisfy the given condition. - Emeric Deutsch, Jun 04 2003 [For a(5) = 4 we have 2143, 1324, 2134 and 1243. - Jon Perry, Sep 14 2013]
Number of 001-avoiding binary words of length n-3. a(n) is the number of partitions of {1, ..., n-1} into two blocks in which only 1- or 2-strings of consecutive integers can appear in a block and there is at least one 2-string. E.g., a(6) = 7 because the enumerated partitions of {1, 2, 3, 4, 5} are 124/35, 134/25, 14/235, 13/245, 1245/3, 145/23, 125/34. - Augustine O. Munagi, Apr 11 2005
Numbers for which only one Fibonacci bit-representation is possible and for which the maximal and minimal Fibonacci bit-representations (A104326 and A014417) are equal. For example, a(12) = 10101 because 8 + 3 + 1 = 12. - Casey Mongoven, Mar 19 2006
Beginning with a(2), the "Recamán transform" (see A005132) of the Fibonacci numbers (A000045). - Nick Hobson, Mar 01 2007
Starting with nonzero terms, a(n) gives the row sums of triangle A158950. - Gary W. Adamson, Mar 31 2009
a(n+2) is the minimum number of elements in an AVL tree of height n. - Lennert Buytenhek (buytenh(AT)wantstofly.org), May 31 2010
a(n) is the number of branch nodes in the Fibonacci tree of order n-1. A Fibonacci tree of order n (n >= 2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node (see the Knuth reference, p. 417). - Emeric Deutsch, Jun 14 2010
a(n+3) is the number of distinct three-strand positive braids of length n (cf. Burckel). - Maxime Bourrigan, Apr 04 2011
a(n+1) is the number of compositions of n with maximal part 2. - Joerg Arndt, May 21 2013
a(n+2) is the number of leafs of great-grandparent DAG (directed acyclic graph) of height n. A great-grandparent DAG of height n is a single node for n = 1; for n > 1 each leaf of ggpDAG(n-1) has two child nodes where pairs of adjacent new nodes are merged into single node if and only if they have disjoint grandparents and same great-grandparent. Consequence: a(n) = 2*a(n-1) - a(n-3). - Hermann Stamm-Wilbrandt, Jul 06 2014
2 and 7 are the only prime numbers in this sequence. - Emmanuel Vantieghem, Oct 01 2014
From Russell Jay Hendel, Mar 15 2015: (Start)
We can establish Gerald McGarvey's conjecture mentioned in the Formula section, however we require n > 4. We need the following 4 prerequisites.
(1) a(n) = F(n) - 1, with {F(n)}A000045.%20(2)%20(Binet%20form)%20F(n)%20=%20(d%5En%20-%20e%5En)/sqrt(5)%20with%20d%20=%20phi%20and%20e%20=%201%20-%20phi,%20de%20=%20-1%20and%20d%20+%20e%20=%201.%20It%20follows%20that%20a(n)%20=%20(d(n)%20-%20e(n))/sqrt(5)%20-%201.%20(3)%20To%20prove%20floor(x)%20=%20y%20is%20equivalent%20to%20proving%20that%20x%20-%20y%20lies%20in%20the%20half-open%20interval%20%5B0,%201).%20(4)%20The%20series%20%7Bs(n)%20=%20c1%20x%5En%20+%20c2%7D">{n >= 1} the Fibonacci numbers A000045. (2) (Binet form) F(n) = (d^n - e^n)/sqrt(5) with d = phi and e = 1 - phi, de = -1 and d + e = 1. It follows that a(n) = (d(n) - e(n))/sqrt(5) - 1. (3) To prove floor(x) = y is equivalent to proving that x - y lies in the half-open interval [0, 1). (4) The series {s(n) = c1 x^n + c2}{n >= 1}, with -1 < x < 0, and c1 and c2 positive constants, converges by oscillation with s(1) < s(3) < s(5) < ... < s(6) < s(4) < s(2). If follows that for any odd n, the open interval (s(n), s(n+1)) contains the subsequence {s(t)}_{t >= n + 2}. Using these prerequisites we can analyze the conjecture.
Using prerequisites (2) and (3) we see we must prove, for all n > 4, that d((d^(n-1) - e^(n-1))/sqrt(5) - 1) - (d^n - e^n)/sqrt(5) + 1 + c lies in the interval [0, 1). But de = -1, implying de^(n-1) = -e^(n-2). It follows that we must equivalently prove (for all n > 4) that E(n, c) = (e^(n-2) + e^n)/sqrt(5) + 1 - d + c = e^(n-2) (e^2 + 1)/sqrt(5) + e + c lies in [0, 1). Clearly, for any particular n, E(n, c) has extrema (maxima, minima) when c = 2*(1-d) and c = (1+d)*(1-d). Therefore, the proof is completed by using prerequisite (4). It suffices to verify E(5, 2*(1-d)) = 0, E(6, 2*(1-d)) = 0.236068, E(5, (1-d)*(1+d)) = 0.618034, E(6, (1-d)*(1+d)) = 0.854102, all lie in [0, 1).
(End)
a(n) can be shown to be the number of distinct nonempty matchings on a path with n vertices. (A matching is a collection of disjoint edges.) - Andrew Penland, Feb 14 2017
Also, for n > 3, the lexicographically earliest sequence of positive integers such that {phi*a(n)} is located strictly between {phi*a(n-1)} and {phi*a(n-2)}. - Ivan Neretin, Mar 23 2017
From Eric M. Schmidt, Jul 17 2017: (Start)
Number of sequences (e(1), ..., e(n-2)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) <= e(k). [Martinez and Savage, 2.5]
Number of sequences (e(1), ..., e(n-2)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) >= e(j) <= e(k) and e(i) != e(k). [Martinez and Savage, 2.5]
(End)
Numbers whose Zeckendorf (A014417) and dual Zeckendorf (A104326) representations are the same: alternating digits of 1 and 0. - Amiram Eldar, Nov 01 2019
a(n+2) is the length of the longest array whose local maximum element can be found in at most n reveals. See link to the puzzle by Alexander S. Kulikov. - Dmitry Kamenetsky, Aug 08 2020
a(n+2) is the number of nonempty subsets of {1,2,...,n} that contain no consecutive elements. For example, the a(6)=7 subsets of {1,2,3,4} are {1}, {2}, {3}, {4}, {1,3}, {1,4} and {2,4}. - Muge Olucoglu, Mar 21 2021
a(n+3) is the number of allowed patterns of length n in the even shift (that is, a(n+3) is the number of binary words of length n in which there are an even number of 0s between any two occurrences of 1). For example, a(7)=12 and the 12 allowed patterns of length 4 in the even shift are 0000, 0001, 0010, 0011, 0100, 0110, 0111, 1000, 1001, 1100, 1110, 1111. - Zoran Sunic, Apr 06 2022
Conjecture: for k a positive odd integer, the sequence {a(k^n): n >= 1} is a strong divisibility sequence; that is, for n, m >= 1, gcd(a(k^n), a(k^m)) = a(k^gcd(n,m)). - Peter Bala, Dec 05 2022
In general, the sum of a second-order linear recurrence having signature (c,d) will be a third-order recurrence having a signature (c+1,d-c,-d). - Gary Detlefs, Jan 05 2023
a(n) is the number of binary strings of length n-2 whose longest run of 1's is of length 1, for n >= 3. - Félix Balado, Apr 03 2025

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 1.
GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 28.
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 64.
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 155.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 31 (1991), 21-29.

Christian G. Bower, Table of n, a(n) for n = 1..500
Isha Agarwal, Matvey Borodin, Aidan Duncan, Kaylee Ji, Tanya Khovanova, Shane Lee, Boyan Litchev, Anshul Rastogi, Garima Rastogi, and Andrew Zhao, From Unequal Chance to a Coin Game Dance: Variants of Penney's Game, arXiv:2006.13002 [math.HO], 2020.
Ricardo Gómez Aíza, Symbolic dynamical scales: modes, orbitals, and transversals, arXiv:2009.02669 [math.DS], 2020.
Kassie Archer and Noel Bourne, Pattern avoidance in compositions and powers of permutations, arXiv:2505.05218 [math.CO], 2025. See pp. 6-7.
Kassie Archer and Aaron Geary, Powers of permutations that avoid chains of patterns, arXiv:2312.14351 [math.CO], 2023. See p. 15.
Mohammad K. Azarian, The Generating Function for the Fibonacci Sequence, Missouri Journal of Mathematical Sciences, Vol. 2, No. 2, Spring 1990, pp. 78-79. Zentralblatt MATH, Zbl 1097.11516.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17.
J.-L. Baril and J.-M. Pallo, Motzkin subposet and Motzkin geodesics in Tamari lattices, 2013.
Erik Bates, Blan Morrison, Mason Rogers, Arianna Serafini, and Anav Sood, A new combinatorial interpretation of partial sums of m-step Fibonacci numbers, arXiv:2503.11055 [math.CO], 2025. See pp. 1-2.
Andrew M. Baxter and Lara K. Pudwell, Ascent sequences avoiding pairs of patterns, 2014.
Serge Burckel, Syntactical methods for braids of three strands, J. Symbolic Comput. 31 (2001), no. 5, 557-564.
Alexander Burstein and Toufik Mansour, Counting occurrences of some subword patterns, arXiv:math/0204320 [math.CO], 2002-2003.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Fan Chung and R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194.
Ligia Loretta Cristea, Ivica Martinjak, and Igor Urbiha, Hyperfibonacci Sequences and Polytopic Numbers, arXiv:1606.06228 [math.CO], 2016.
Michael Dairyko, Samantha Tyner, Lara Pudwell, and Casey Wynn, Non-contiguous pattern avoidance in binary trees. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From _N. J. A. Sloane_, Feb 01 2013
Emeric Deutsch, Problem Q915, Math. Magazine, vol. 74, No. 5, 2001, p. 404.
Christian Ennis, William Holland, Omer Mujawar, Aadit Narayanan, Frank Neubrander, Marie Neubrander, and Christina Simino, Words in Random Binary Sequences I, arXiv:2107.01029 [math.GM], 2021.
Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. 19 (2024), no. 19, Paper #1, 27 pp.
Fumio Hazama, Spectra of graphs attached to the space of melodies, Discrete Math., 311 (2011), 2368-2383. See Table 2.1.
Yasuichi Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178. [From _Emeric Deutsch_, Jun 14 2010]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 384
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 96.
Scott O. Jones and P. Mark Kayll, Constructing Edge-labellings of K_n, with Constant-length Hamilton Cycles, J. Comb. Math. Comb. Comp. (2006) Vol. 57, pp. 83-95. See p. 92.
Tamara Kogan, L. Sapir, A. Sapir, and A. Sapir, The Fibonacci family of iterative processes for solving nonlinear equations, Applied Numerical Mathematics 110 (2016) 148-158.
Alexander S. Kulikov, Find Local Maximum in an Integer Sequence, Puzzling Stack Exchange, 2020.
René Lagrange, Quelques résultats dans la métrique des permutations, Annales Scientifiques de l'Ecole Normale Supérieure, Paris, 79 (1962), 199-241.
D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
Rui Liu and Feng-Zhen Zhao, On the Sums of Reciprocal Hyperfibonacci Numbers and Hyperlucas Numbers, Journal of Integer Sequences, Vol. 15 (2012), #12.4.5. - From _N. J. A. Sloane_, Oct 05 2012
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
El-Mehdi Mehiri, Saad Mneimneh, and Hacène Belbachir, The Towers of Fibonacci, Lucas, Pell, and Jacobsthal, arXiv:2502.11045 [math.CO], 2025. See p. 12.
Augustine O. Munagi, Set Partitions with Successions and Separations,IJMMS 2005:3 (2005), 451-463.
Sam Northshield, Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,..., Amer. Math. Month., Vol. 117 (7), pp. 581-598, 2010.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Lara Pudwell, Pattern avoidance in trees, (slides from a talk, mentions many sequences), 2012.
Lara Pudwell, Pattern-avoiding ascent sequences, Slides from a talk, 2015.
Stacey Wagner, Enumerating Alternating Permutations with One Alternating Descent, DePaul Discoveries: Vol. 2: Iss. 1, Article 2.
Hsin-Po Wang and Chi-Wei Chin, On Counting Subsequences and Higher-Order Fibonacci Numbers, arXiv:2405.17499 [cs.IT], 2024. See p. 2.
Arthur T. White, Ringing the changes, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 203-215.
Peijun Xu, Growth of positive braids semigroups, Journal of Pure and Applied Algebra, 1992.
J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 31 (1991), 21-29. (Annotated scanned copy)
Jianqiang Zhao, Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, arXiv preprint arXiv:1412.8044 [math.NT], 2014. See Table 9, line 1.
Li-Na Zheng, Rui Liu, and Feng-Zhen Zhao, On the Log-Concavity of the Hyperfibonacci Numbers and the Hyperlucas Numbers, Journal of Integer Sequences, Vol. 17 (2014), #14.1.4.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000032, A000045, A000119, A001611, A001654, A001911, A005968, A005969.
Cf. A006327, A014417, A054761, A081006, A081007, A081008, A081009, A083368.
Cf. A098531, A098532, A098533, A101220, A104326, A105488, A105489, A119282.
Cf. A128697, A157725, A157726, A157727, A157728, A157729, A158950, A167616.
Cf. A001622, A228074.
Antidiagonal sums of array A004070.
Right-hand column 2 of triangle A011794.
Related to sum of Fibonacci(kn) over n. Cf. A099919, A058038, A138134, A053606.
Subsequence of A226538. Also a subsequence of A061489.

a000071 n = a000071_list !! n
a000071_list = map (subtract 1) $ tail a000045_list
-- Reinhard Zumkeller, May 23 2013

[Fibonacci(n)-1: n in [1..60]]; // Vincenzo Librandi, Apr 04 2011

A000071 := proc(n) combinat[fibonacci](n)-1 ; end proc; # R. J. Mathar, Apr 07 2011
a:= n-> (Matrix([[1, 1, 0], [1, 0, 0], [1, 0, 1]])^(n-1))[3, 2]; seq(a(n), n=1..50); # Alois P. Heinz, Jul 24 2008

Fibonacci[Range[40]] - 1 (* or *) LinearRecurrence[{2, 0, -1}, {0, 0, 1}, 40] (* Harvey P. Dale, Aug 23 2013 *)
Join[{0}, Accumulate[Fibonacci[Range[0, 39]]]] (* Alonso del Arte, Oct 22 2017, based on Giorgi Dalakishvili's formula *)

PARI
```
{a(n) = if( n<1, 0, fibonacci(n)-1)};
```

[fibonacci(n)-1 for n in range(1,60)] # G. C. Greubel, Oct 21 2024

a(n) = A000045(n) - 1.
a(0) = -1, a(1) = 0; thereafter a(n) = a(n-1) + a(n-2) + 1.
a(n) = A101220(1, 1, n-2), for n > 1.
G.f.: x^3/((1-x-x^2)*(1-x)). - Simon Plouffe in his 1992 dissertation, dropping initial 0's
a(n) = 2*a(n-1) - a(n-3). - R. H. Hardin, Apr 02 2011
Partial sums of Fibonacci numbers. - Wolfdieter Lang
a(n) = -1 + (A*B^n + C*D^n)/10, with A, C = 5 +- 3*sqrt(5), B, D = (1 +- sqrt(5))/2. - Ralf Stephan, Mar 02 2003
a(1) = 0, a(2) = 0, a(3) = 1, then a(n) = ceiling(phi*a(n-1)) where phi is the golden ratio (1 + sqrt(5))/2. - Benoit Cloitre, May 06 2003
Conjecture: for all c such that 2*(2 - Phi) <= c < (2 + Phi)*(2 - Phi) we have a(n) = floor(Phi*a(n-1) + c) for n > 4. - Gerald McGarvey, Jul 22 2004. This is true provided n > 3 is changed to n > 4, see proof in Comments section. - Russell Jay Hendel, Mar 15 2015
a(n) = Sum_{k = 0..floor((n-2)/2)} binomial(n-k-2, k+1). - Paul Barry, Sep 23 2004
a(n+3) = Sum_{k = 0..floor(n/3)} binomial(n-2*k, k)*(-1)^k*2^(n-3*k). - Paul Barry, Oct 20 2004
a(n+1) = Sum(binomial(n-r, r)), r = 1, 2, ... which is the case t = 2 and k = 2 in the general case of t-strings and k blocks: a(n+1, k, t) = Sum(binomial(n-r*(t-1), r)*S2(n-r*(t-1)-1, k-1)), r = 1, 2, ... - Augustine O. Munagi, Apr 11 2005
a(n) = Sum_{k = 0..n-2} k*Fibonacci(n - k - 3). - Ross La Haye, May 31 2006
a(n) = term (3, 2) in the 3 X 3 matrix [1, 1, 0; 1, 0, 0; 1, 0, 1]^(n-1). - Alois P. Heinz, Jul 24 2008
For n >= 4, a(n) = ceiling(phi*a(n-1)), where phi is the golden ratio. - Vladimir Shevelev, Jul 04 2010
Closed-form without two leading zeros g.f.: 1/(1 - 2*x - x^3); ((5 + 2*sqrt(5))*((1 + sqrt(5))/2)^n + (5 - 2*sqrt(5))*((1 - sqrt(5))/2)^n - 5)/5; closed-form with two leading 0's g.f.: x^2/(1 - 2*x - x^3); ((5 + sqrt(5))*((1 + sqrt(5))/2)^n + (5 - sqrt(5))*((1 - sqrt(5))/2)^n - 10)/10. - Tim Monahan, Jul 10 2011
A000119(a(n)) = 1. - Reinhard Zumkeller, Dec 28 2012
a(n) = A228074(n - 1, 2) for n > 2. - Reinhard Zumkeller, Aug 15 2013
G.f.: Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(4*k + 2 - x^2)/( x*(4*k + 4 - x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013
A083368(a(n+3)) = n. - Reinhard Zumkeller, Aug 10 2014
E.g.f.: 1 - exp(x) + 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Ilya Gutkovskiy, Jun 15 2016
a(n) = A000032(3+n) - 1 mod A000045(3+n). - Mario C. Enriquez, Apr 01 2017
a(n) = Sum_{i=0..n-2} Fibonacci(i). - Giorgi Dalakishvili (mcnamara_gio(AT)yahoo.com), Apr 02 2005 [corrected by Doug Bell, Jun 01 2017]
a(n+2) = Sum_{j = 0..floor(n/2)} Sum_{k = 0..j} binomial(n - 2*j, k+1)*binomial(j, k). - Tony Foster III, Sep 08 2017
From Peter Bala, Nov 12 2021: (Start)
a(4*n) = Fibonacci(2*n+1)*Lucas(2*n-1) = A081006(n);
a(4*n+1) = Fibonacci(2*n)*Lucas(2*n+1) = A081007(n);
a(4*n+2) = Fibonacci(2*n)*Lucas(2*n+2) = A081008(n);
a(4*n+3) = Fibonacci(2*n+2)*Lucas(2*n+1) = A081009(n). (End)
G.f.: x^3/((1 - x - x^2)*(1 - x)) = Sum_{n >= 0} (-1)^n * x^(n+3) *( Product_{k = 1..n} (k - x)/Product_{k = 1..n+2} (1 - k*x) )  (a telescoping series). - Peter Bala, May 08 2024
Product_{n>=4} (1 + (-1)^n/a(n)) = 3*phi/4, where phi is the golden ratio (A001622). - Amiram Eldar, Nov 28 2024

Edited by N. J. A. Sloane, Apr 04 2011

A055248 Triangle of partial row sums of triangle A007318(n,m) (Pascal's triangle). Triangle A008949 read backwards. Riordan (1/(1-2x), x/(1-x)).

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 11, 5, 1, 32, 31, 26, 16, 6, 1, 64, 63, 57, 42, 22, 7, 1, 128, 127, 120, 99, 64, 29, 8, 1, 256, 255, 247, 219, 163, 93, 37, 9, 1, 512, 511, 502, 466, 382, 256, 130, 46, 10, 1, 1024, 1023, 1013, 968, 848, 638, 386, 176, 56, 11, 1

Offset: 0

Wolfdieter Lang, May 26 2000

In the language of the Shapiro et al. reference (also given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The g.f. for the row polynomials p(n,x) (increasing powers of x) is 1/((1-2*z)*(1-x*z/(1-z))).
Binomial transform of the all 1's triangle: as a Riordan array, it factors to give (1/(1-x),x/(1-x))(1/(1-x),x). Viewed as a number square read by antidiagonals, it has T(n,k) = Sum_{j=0..n} binomial(n+k,n-j) and is then the binomial transform of the Whitney square A004070. - Paul Barry, Feb 03 2005
Riordan array (1/(1-2x), x/(1-x)). Antidiagonal sums are A027934(n+1), n >= 0. - Paul Barry, Jan 30 2005; edited by Wolfdieter Lang, Jan 09 2015
Eigensequence of the triangle = A005493: (1, 3, 10, 37, 151, 674, ...); row sums of triangles A011971 and A159573. - Gary W. Adamson, Apr 16 2009
Read as a square array, this is the generalized Riordan array ( 1/(1 - 2*x), 1/(1 - x) ) as defined in the Bala link (p. 5), which factorizes as ( 1/(1 - x), x/(1 - x) )*( 1/(1 - x), x )*( 1, 1 + x ) = P*U*transpose(P), where P denotes Pascal's triangle, A007318, and U is the lower unit triangular array with 1's on or below the main diagonal. - Peter Bala, Jan 13 2016

			The triangle a(n,m) begins:
n\m    0    1    2   3   4   5   6   7  8  9 10 ...
0:     1
1:     2    1
2:     4    3    1
3:     8    7    4   1
4:    16   15   11   5   1
5:    32   31   26  16   6   1
6:    64   63   57  42  22   7   1
7:   128  127  120  99  64  29   8   1
8:   256  255  247 219 163  93  37   9  1
9:   512  511  502 466 382 256 130  46 10  1
10: 1024 1023 1013 968 848 638 386 176 56 11  1
... Reformatted. - _Wolfdieter Lang_, Jan 09 2015
Fourth row polynomial (n=3): p(3,x)= 8 + 7*x + 4*x^2 + x^3.
The matrix inverse starts
   1;
  -2,   1;
   2,  -3,   1;
  -2,   5,  -4,    1;
   2,  -7,   9,   -5,    1;
  -2,   9, -16,   14,   -6,    1;
   2, -11,  25,-  30,   20,   -7,    1;
  -2,  13, -36,   55,  -50,   27,   -8,    1;
   2, -15,  49,  -91,  105,  -77,   35,   -9,  1;
  -2,  17, -64,  140, -196,  182, -112,   44, -10,   1;
   2, -19,  81, -204,  336, -378,  294, -156,  54, -11, 1;
   ...
which may be related to A029653. - _R. J. Mathar_, Mar 29 2013
From _Peter Bala_, Dec 23 2014: (Start)
With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins
/1      \ /1        \ /1       \       /1       \
|2 1     ||0 1       ||0 1      |      |2  1     |
|4 3 1   ||0 2 1     ||0 0 1    |... = |4  5 1   |
|8 7 4 1 ||0 4 3 1   ||0 0 2 1  |      |8 19 9 1 |
|...     ||0 8 7 4 1 ||0 0 4 3 1|      |...      |
|...     ||...       ||...      |      |         |
= A143494. (End)
Matrix factorization of square array as P*U*transpose(P):
/1      \ /1        \ /1 1 1 1 ...\    /1  1  1  1 ...\
|1 1     ||1 1       ||0 1 2 3 ... |   |2  3  4  5 ... |
|1 2 1   ||1 1 1     ||0 0 1 3 ... | = |4  7 11 16 ... |
|1 3 3 1 ||1 1 1 1   ||0 0 0 1 ... |   |8 15 26 42 ... |
|...     ||...       ||...         |   |...            |
- _Peter Bala_, Jan 13 2016

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Peter Bala, Notes on generalized Riordan arrays
Peter Bala, A055248: Rapidly converging series for log(2) and Pi
Jean-Luc Baril, Javier F. González, and José L. Ramírez, Last symbol distribution in pattern avoiding Catalan words, Univ. Bourgogne (France, 2022).
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Norman Lindquist and Gerard Sierksma, Extensions of set partitions, Journal of Combinatorial Theory, Series A 31.2 (1981): 190-198. See Table I.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

Column sequences: A000079 (powers of 2, m=0), A000225 (m=1), A000295 (m=2), A002662 (m=3), A002663 (m=4), A002664 (m=5), A035038 (m=6), A035039 (m=7), A035040 (m=8), A035041 (m=9), A035042 (m=10).
Row sums: A001792(n) = A055249(n, 0).
Alternating row sums: A011782.
Cf. A011971, A159573. - Gary W. Adamson, Apr 16 2009
Cf. A007318, A008949, A106516, A143494.

a055248 n k = a055248_tabl !! n !! k
a055248_row n = a055248_tabl !! n
a055248_tabl = map reverse a008949_tabl
-- Reinhard Zumkeller, Jun 20 2015

T := (n,k) -> 2^n - (1/2)*binomial(n, k-1)*hypergeom([1, n + 1], [n-k + 2], 1/2).
seq(seq(simplify(T(n,k)), k=0..n),n=0..10); # Peter Luschny, Oct 10 2019

a[n_, m_] := Sum[ Binomial[n, m + j], {j, 0, n}]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Paul Barry *)
T[n_, k_] := Binomial[n, k] * Hypergeometric2F1[1, k - n, k + 1, -1];
Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* Peter Luschny, Oct 06 2023 *)

a(n, m) = A008949(n, n-m), if n > m >= 0.
a(n, m) = Sum_{k=m..n} A007318(n, k) (partial row sums in columns m).
Column m recursion: a(n, m) = Sum_{j=m..n-1} a(j, m) + A007318(n, m) if n >= m >= 0, a(n, m) := 0 if nG.f. for column m: (1/(1-2*x))*(x/(1-x))^m, m >= 0.
a(n, m) = Sum_{j=0..n} binomial(n, m+j). - Paul Barry, Feb 03 2005
Inverse binomial transform (by columns) of A112626. - Ross La Haye, Dec 31 2006
T(2n,n) = A032443(n). - Philippe Deléham, Sep 16 2009
From Peter Bala, Dec 23 2014: (Start)
Exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(8 + 7*x + 4*x^2/2! + x^3/3!) = 8 + 15*x + 26*x^2/2! + 42*x^3/3! + 64*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ).
Let M denote the present triangle. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A143494 (but with a different offset). See the Example section. Cf. A106516. (End)
a(n,m) = Sum_{p=m..n} 2^(n-p)*binomial(p-1,m-1), n >= m >= 0, else 0. - Wolfdieter Lang, Jan 09 2015
T(n, k) = 2^n - (1/2)*binomial(n, k-1)*hypergeom([1, n+1], [n-k+2], 1/2). - Peter Luschny, Oct 10 2019
T(n, k) = binomial(n, k)*hypergeom([1, k - n], [k + 1], -1). - Peter Luschny, Oct 06 2023
n-th row polynomial R(n, x) = (2^n - x*(1 + x)^n)/(1 - x). These polynomials can be used to find series acceleration formulas for the constants log(2) and Pi. - Peter Bala, Mar 03 2025

A052509 Knights-move Pascal triangle: T(n,k), n >= 0, 0 <= k <= n; T(n,0) = T(n,n) = 1, T(n,k) = T(n-1,k) + T(n-2,k-1) for k = 1,2,...,n-1, n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 7, 4, 2, 1, 1, 6, 11, 8, 4, 2, 1, 1, 7, 16, 15, 8, 4, 2, 1, 1, 8, 22, 26, 16, 8, 4, 2, 1, 1, 9, 29, 42, 31, 16, 8, 4, 2, 1, 1, 10, 37, 64, 57, 32, 16, 8, 4, 2, 1, 1, 11, 46, 93, 99, 63, 32, 16, 8, 4, 2, 1

Offset: 0

N. J. A. Sloane, Mar 17 2000

Also square array T(n,k) (n >= 0, k >= 0) read by antidiagonals: T(n,k) = Sum_{i=0..k} binomial(n,i).
As a number triangle read by rows, this is T(n,k) = Sum_{i=n-2*k..n-k} binomial(n-k,i), with T(n,k) = T(n-1,k) + T(n-2,k-1). Row sums are A000071(n+2). Diagonal sums are A023435(n+1). It is the reverse of the Whitney triangle A004070. - Paul Barry, Sep 04 2005
Also, twice number of orthants intersected by a generic k-dimensional subspace of R^n [Naiman and Scheinerman, 2017]. - N. J. A. Sloane, Mar 03 2018

			Triangle begins:
[0] 1;
[1] 1, 1;
[2] 1, 2,  1;
[3] 1, 3,  2,  1;
[4] 1, 4,  4,  2,  1;
[5] 1, 5,  7,  4,  2,  1;
[6] 1, 6, 11,  8,  4,  2, 1;
[7] 1, 7, 16, 15,  8,  4, 2, 1;
[8] 1, 8, 22, 26, 16,  8, 4, 2, 1;
[9] 1, 9, 29, 42, 31, 16, 8, 4, 2, 1;
As a square array, this begins:
  1  1  1  1  1  1 ...
  1  2  2  2  2  2 ...
  1  3  4  4  4  4 ...
  1  4  7  8  8  8 ...
  1  5 11 15 16 ...
  1  6 16 26 31 32 ...

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 1C.
Daniel Q. Naiman and Edward R. Scheinerman, Arbitrage and Geometry, arXiv:1709.07446 [q-fin.MF], 2017 [Contains the square array multiplied by 2].
Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See Tables 5 and 14.
D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A054123, A054124, A007318, A008949.
Row sums A000071; Diagonal sums A023435; Mirror A004070.
Columns give A000027, A000124, A000125, A000127, A006261, ...
Cf. A052509, A054123, A054124, A007318, A008949, A052553.
Partial sums across rows of (extended) Pascal's triangle A052553.

A052509:=Flat(List([0..100],n->List([0..n],k->Sum([0..n],m->Binomial(n-k,k-m))))); # Muniru A Asiru, Sat Feb 17 2018

a052509 n k = a052509_tabl !! n !! k
a052509_row n = a052509_tabl !! n
a052509_tabl = [1] : [1,1] : f [1] [1,1] where
   f row' row = rs : f row rs where
     rs = zipWith (+) ([0] ++ row' ++ [1]) (row ++ [0])
-- Reinhard Zumkeller, Nov 22 2012

[[(&+[Binomial(n-k, k-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 13 2019

a := proc(n::nonnegint, k::nonnegint) option remember: if k=0 then RETURN(1) fi: if k=n then RETURN(1) fi: a(n-1,k)+a(n-2,k-1) end: for n from 0 to 11 do for k from 0 to n do printf(`%d,`,a(n,k)) od: od: # James Sellers, Mar 17 2000
with(combinat): for s from 0 to 11 do for n from s to 0 by -1 do if n=0 or s-n=0 then printf(`%d,`,1) else printf(`%d,`,sum(binomial(n, i), i=0..s-n)) fi; od: od: # James Sellers, Mar 17 2000

Table[Sum[Binomial[n-k, k-m], {m, 0, n}], {n, 0, 10}, {k, 0, n}]
T[n_, k_] := Hypergeometric2F1[-k, -n + k, -k, -1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Nov 28 2021 *)

T(n,k)=sum(m=0,n,binomial(n-k,k-m));
for(n=0,10,for(k=0,n,print1(T(n,k),", "););print();); /* show triangle */

[[sum(binomial(n-k, k-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 13 2019

T(n, k) = Sum_{m=0..n} binomial(n-k, k-m). - Wouter Meeussen, Oct 03 2002
From Werner Schulte, Feb 15 2018: (Start)
Referring to the square array T(i,j):
G.f. of row n: Sum_{k>=0} T(n,k) * x^k = (1+x)^n / (1-x).
G.f. of T(i,j): Sum_{k>=0, n>=0} T(n,k) * x^k * y^n = 1 / ((1-x)*(1-y-x*y)).
Let a_i(n) be multiplicative with a_i(p^e) = T(i, e), p prime and e >= 0, then Sum_{n>0} a_i(n)/n^s = (zeta(s))^(i+1) / (zeta(2*s))^i for i >= 0.
(End)
T(n, k) = hypergeom([-k, -n + k], [-k], -1). - Peter Luschny, Nov 28 2021
From Jianing Song, May 30 2022: (Start)
Referring to the triangle, G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = 1 / ((1-x*y)*(1-x-x^2*y)).
T(n,k) = 2^(n-k) for ceiling(n/2) <= k <= n. (End)

More terms from James Sellers, Mar 17 2000
Entry formed by merging two earlier entries. - N. J. A. Sloane, Jun 17 2007
Edited by Johannes W. Meijer, Jul 24 2011

A067331 Convolution of Fibonacci F(n+1), n >= 0, with F(n+3), n >= 0.

Original entry on oeis.org

2, 5, 12, 25, 50, 96, 180, 331, 600, 1075, 1908, 3360, 5878, 10225, 17700, 30509, 52390, 89664, 153000, 260375, 442032, 748775, 1265832, 2136000, 3598250, 6052061, 10164540, 17048641, 28559450, 47786400, 79870428, 133359715, 222457608, 370747675, 617363100

Offset: 0

Wolfdieter Lang, Feb 15 2002

Third diagonal of A067330. Third column of A067418.
From Emeric Deutsch, Jun 15 2010: (Start)
a(n) is the external path length of the Fibonacci tree of order n+3. A Fibonacci tree of order n (n >= 2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. The external path length of a tree is the sum of the levels of its external nodes (i.e., leaves).
a(n) = Sum_{k>=0} k*A178524(n+2,k).
(End)
a(n) equals the penultimate immanant of the (n+3) X (n+3) tridiagonal matrix with ones along the main diagonal, the superdiagonal, and the subdiagonal. - John M. Campbell, Jan 01 2016
a(n) is the sum of the eccentricities of the vertices of the Fibonacci cube G(n+1). Example: a(1)=5; indeed, the Fibonacci cube G(2) is the path graph P(3), the vertices of which have eccentricities 2, 1, 2. - Emeric Deutsch, May 28 2017

			From _John M. Campbell_, Jan 03 2016: (Start)
Letting n=2, the external path length of the Fibonacci tree T(5) of order n+3=5 illustrated below is 12 = a(2) = F(1)*F(5) + F(2)*F(4) + F(3)*F(3).
     .
    / \
   /\ /\
  /\
(End)

D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

Robert Israel, Table of n, a(n) for n = 0..4720
Matthew Blair, Rigoberto Flórez, Antara Mukherjee, and José L. Ramírez, Matrices in the determinant Hosoya triangle, Fibonacci Quart. 58 (2020), no. 5, 34-54.
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Geometric Patterns in The Determinant Hosoya Triangle, INTEGERS, A90, 2021.
J. Bodeen, S. Butler, T. Kim, X. Sun, and S. Wang, Tiling a strip with triangles, Electron. J. Combin. 21 (1) (2014), P1.7.
John M. Campbell, On the external path length of a Fibonacci tree.
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20(2) (1982), 168-178.
S. Klavzar and M. Mollard, Asymptotic properties of Fibonacci cubes and Lucas cubes, HAL Id: hal-00836788, 2013.
S. Klavzar and M. Mollard, Asymptotic properties of Fibonacci cubes and Lucas cubes, Ann. Comb. 18 (2014), 447-457.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Row sums of A108038.

Cf. A000045, A004070, A067330, A067418, A178523, A178524.

[((7*n+10)*Fibonacci(n+1)+4*(n+1)*Fibonacci(n))/5: n in [0..40]]; // Vincenzo Librandi, Jan 02 2016

f:= gfun:-rectoproc({a(n) = 2*a(n-1)+a(n-2) - 2*a(n-3)-a(n-4),a(0)=2,a(1)=5,a(2)=12,a(3)=25},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jan 06 2016

LinearRecurrence[{2, 1, -2, -1}, {2, 5, 12, 25}, 70] (* Vincenzo Librandi, Jan 02 2016 *)
Table[SeriesCoefficient[(2 + x)/(1 - x - x^2)^2, {x, 0, n}], {n, 0, 34}] (* Michael De Vlieger, Jan 02 2016 *)
Print[Table[Sum[Binomial[n + 3 - i, i]*(n + 2 - 2*i), {i, 0, Floor[(n + 3)/2]}], {n, 0, 100}]] (* John M. Campbell, Jan 04 2016 *)
Module[{nn=40,fibs},fibs=Fibonacci[Range[nn]];Table[ListConvolve[Take[ fibs,n],Take[fibs,{2,n+2}]],{n,nn-2}]][[All,2]] (* Harvey P. Dale, Aug 03 2019 *)

Vec((2+x)/(1-x-x^2)^2 + O(x^100)) \\ Altug Alkan, Jan 04 2016

a(n) = A067330(n+2, n) = A067418(n+2, 2) = Sum_{k=0..n} F(k+1)*F(n+3-k), n >= 0.
a(n) = ((7*n + 10)*F(n + 1) + 4*(n + 1)*F(n))/5, with F(n) = A000045(n) (Fibonacci).
G.f.: (2 + x)/(1 - x - x^2)^2.
a(n) = Sum_{i=0..floor((n+3)/2)} binomial(n+3-i, i)*(n + 2 - 2*i). - John M. Campbell, Jan 04 2016
E.g.f.: exp(x/2)*((50 + 55*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(18 + 25*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 04 2023

Showing 1-10 of 26 results. Next

cp's OEIS Frontend

A097750 Reversal of the binomial transform of the Whitney triangle A004070 (see A131250), triangle read by rows, T(n,k) for 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A131248 2*A004070 - A000012.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A131250 A007318 * A004070.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A131254 A004070 * A000012.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A131255 A004070 * A000012(signed).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A001045 Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer to 2^n/3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

PARI

PARI

PARI

Python

Python

Sage

Formula

Extensions

A000071 a(n) = Fibonacci(n) - 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell