A014551 -id:A014551 - 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.
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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

A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane

Least significant bit of n, lsb(n).
Also decimal expansion of 1/99.
Also the binary expansion of 1/3. - Robert G. Wilson v, Sep 01 2015
a(n) = A134451(n) mod 2. - Reinhard Zumkeller, Oct 27 2007 [Corrected by Jianing Song, Nov 22 2019]
Characteristic function of odd numbers: a(A005408(n)) = 1, a(A005843(n)) = 0. - Reinhard Zumkeller, Sep 29 2008
A102370(n) modulo 2. - Philippe Deléham, Apr 04 2009
Base b expansion of 1/(b^2-1) for any b >= 2 is 0.0101... (A005563 has b^2-1). - Rick L. Shepherd, Sep 27 2009
Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 2, A[i,i] := 1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,1). - Milan Janjic, Jan 24 2010
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the principal Dirichlet character of the reduced residue system mod 2 or mod 4 or mod 8 or mod 16 ...
Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 == A111003,
or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.05179979... = 7*A002117/8,
or L(4,chi) = Sum_{n>=1} a(n)/n^4 = 1.014678... = A092425/96. (End)
Also parity of the nonnegative integers A001477. - Omar E. Pol, Jan 17 2012
a(n) = (4/n), where (k/n) is the Kronecker symbol. See the Eric Weisstein link. - Wolfdieter Lang, May 28 2013
Also the inverse binomial transform of A131577. - Paul Curtz, Nov 16 2016 [an observation forwarded by Jean-François Alcover]
The emanation sequence for the globe category. That is take the globe category, take the corresponding polynomial comonad, consider its carrier polynomial as a generating function, and take the corresponding sequence. - David Spivak, Sep 25 2020
For n > 0, a(n) is the alternating sum of the product of n increasing and n decreasing odd factors. For example, a(4) = 1*7 - 3*5 + 5*3 - 7*1 and a(5) = 1*9 - 3*7 + 5*5 - 7*3 + 9*1. - Charlie Marion, Mar 24 2022

			G.f. = x + x^3 + x^5 + x^7 + x^9 + x^11 + x^13 + x^15 + ... - _Michael Somos_, Feb 20 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Wasserman, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function
Eric Weisstein's World of Mathematics, Kronecker Symbol
A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29
Index entries for "core" sequences
Index entries for sequences that are fixed points of mappings
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,1).

Ones complement of A059841.
Cf. A053644 for most significant bit.
This is Guy Steele's sequence GS(1, 2) (see A135416).
Period k zigzag sequences: this sequence (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).
Cf. A154955 (Mobius transform), A131577 (binomial transform).
Cf. A111003 (Dgf at s=2), A233091 (Dgf at s=3), A300707 (Dgf at s=4).
Parity of A005811.

a000035 n = n `mod` 2  -- James Spahlinger, Oct 08 2012

a000035_list = cycle [0,1]  -- Reinhard Zumkeller, Jan 06 2012

A000035 := n->n mod 2;
[ seq(i mod 2, i=0..100) ];

PadLeft[{},110,{0,1}] (* Harvey P. Dale, Sep 25 2011 *)

A000035(n):=mod(n,2)$
makelist(A000035(n),n,0,30); /* Martin Ettl, Nov 12 2012 */

PARI
```
a(n)=n%2;
```

a(n)=direuler(p=1,100,if(p==2,1,1/(1-X)))[n] /* Ralf Stephan, Mar 27 2015 */

def A000035(n): return n & 1 # Chai Wah Wu, May 25 2022

(define (A000035 n) (mod n 2)) ;; For R6RS. Use modulo in older Schemes like MIT/GNU Scheme. - Antti Karttunen, Mar 21 2017

a(n) = (1 - (-1)^n)/2.
a(n) = n mod 2.
a(n) = 1 - a(n-1).
Multiplicative with a(p^e) = p mod 2. - David W. Wilson, Aug 01 2001
G.f.: x/(1-x^2). E.g.f.: sinh(x). - Paul Barry, Mar 11 2003
a(n) = (A000051(n) - A014551(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2003
a(n) = ceiling((-2)^(-n-1)). - Reinhard Zumkeller, Apr 19 2005
Dirichlet g.f.: (1-1/2^s)*zeta(s). - R. J. Mathar, Mar 04 2011
a(n) = ceiling(n/2) - floor(n/2). - Arkadiusz Wesolowski, Sep 16 2012
a(n) = ceiling( cos(Pi*(n-1))/2 ). - Wesley Ivan Hurt, Jun 16 2013
a(n) = floor((n-1)/2) - floor((n-2)/2). - Mikael Aaltonen, Feb 26 2015
Dirichlet g.f.: L(chi(2),s) with chi(2) the principal Dirichlet character modulo 2. - Ralf Stephan, Mar 27 2015
a(n) = 0^^n = 0^(0^(0...)) (n times), where we take 0^0 to be 1. - Natan Arie Consigli, May 02 2015
Euler transform and inverse Moebius transform of length 2 sequence [0, 1]. - Michael Somos, Feb 20 2024

A002450 a(n) = (4^n - 1)/3.

Original entry on oeis.org

0, 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, 349525, 1398101, 5592405, 22369621, 89478485, 357913941, 1431655765, 5726623061, 22906492245, 91625968981, 366503875925, 1466015503701, 5864062014805, 23456248059221, 93824992236885, 375299968947541

Offset: 0

N. J. A. Sloane

For n > 0, a(n) is the degree (n-1) "numbral" power of 5 (see A048888 for the definition of numbral arithmetic). Example: a(3) = 21, since the numbral square of 5 is 5(*)5 = 101(*)101(base 2) = 101 OR 10100 = 10101(base 2) = 21, where the OR is taken bitwise. - John W. Layman, Dec 18 2001
a(n) is composite for all n > 2 and has factors x, (3*x + 2*(-1)^n) where x belongs to A001045. In binary the terms greater than 0 are 1, 101, 10101, 1010101, etc. - John McNamara, Jan 16 2002
Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - R. H. Hardin, Mar 16 2002
The Collatz-function iteration started at a(n), for n >= 1, will end at 1 after 2*n+1 steps. - Labos Elemer, Sep 30 2002 [corrected by Wolfdieter Lang, Aug 16 2021]
Second binomial transform of A001045. - Paul Barry, Mar 28 2003
All members of sequence are also generalized octagonal numbers (A001082). - Matthew Vandermast, Apr 10 2003
Also sum of squares of divisors of 2^(n-1): a(n) = A001157(A000079(n-1)), for n > 0. - Paul Barry, Apr 11 2003
Binomial transform of A000244 (with leading zero). - Paul Barry, Apr 11 2003
Number of walks of length 2n between two vertices at distance 2 in the cycle graph C_6. For n = 2 we have for example 5 walks of length 4 from vertex A to C: ABABC, ABCBC, ABCDC, AFABC and AFEDC. - Herbert Kociemba, May 31 2004
Also number of walks of length 2n + 1 between two vertices at distance 3 in the cycle graph C_12. - Herbert Kociemba, Jul 05 2004
a(n+1) is the number of steps that are made when generating all n-step random walks that begin in a given point P on a two-dimensional square lattice. To make one step means to mark one vertex on the lattice (compare A080674). - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Mar 13 2005
a(n+1) is the sum of square divisors of 4^n. - Paul Barry, Oct 13 2005
a(n+1) is the decimal number generated by the binary bits in the n-th generation of the Rule 250 elementary cellular automaton. - Eric W. Weisstein, Apr 08 2006
a(k) = [M^k]2,1, where M is the 3 X 3 matrix defined as follows: M = [1, 1, 1; 1, 3, 1; 1, 1, 1]. - _Simone Severini, Jun 11 2006
a(n-1) / a(n) = percentage of wasted storage if a single image is stored as a pyramid with a each subsequent higher resolution layer containing four times as many pixels as the previous layer. n is the number of layers. - Victor Brodsky (victorbrodsky(AT)gmail.com), Jun 15 2006
k is in the sequence if and only if C(4k + 1, k) (A052203) is odd. - Paul Barry, Mar 26 2007
This sequence also gives the number of distinct 3-colorings of the odd cycle C(2*n - 1). - Keith Briggs, Jun 19 2007
All numbers of the form m*4^m + (4^m-1)/3 have the property that they are sums of two squares and also their indices are the sum of two squares. This follows from the identity m*4^m + (4^m-1)/3 = 4(4(..4(4m + 1) + 1) + 1) + 1 ..) + 1. - Artur Jasinski, Nov 12 2007
For n > 0, terms are the numbers that, in base 4, are repunits: 1_4, 11_4, 111_4, 1111_4, etc. - Artur Jasinski, Sep 30 2008
Let A be the Hessenberg matrix of order n, defined by: A[1, j] = 1, A[i, i] := 5, (i > 1), A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 1, a(n) = charpoly(A,1). - Milan Janjic, Jan 27 2010
This is the sequence A(0, 1; 3, 4; 2) = A(0, 1; 4, 0; 1) of the family of sequences [a, b : c, d : k] considered by G. Detlefs, and treated as A(a, b; c, d; k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
6*a(n) + 1 is every second Mersenne number greater than or equal to M3, hence all Mersenne primes greater than M2 must be a 6*a(n) + 1 of this sequence. - Roderick MacPhee, Nov 01 2010
Smallest number having alternating bit sum n. Cf. A065359.
For n = 1, 2, ..., the last digit of a(n) is 1, 5, 1, 5, ... . - Washington Bomfim, Jan 21 2011
Rule 50 elementary cellular automaton generates this sequence. This sequence also appears in the second column of array in A173588. - Paul Muljadi, Jan 27 2011
Sequence found by reading the line from 0, in the direction 0, 5, ... and the line from 1, in the direction 1, 21, ..., in the square spiral whose edges are the Jacobsthal numbers A001045 and whose vertices are the numbers A000975. These parallel lines are two semi-diagonals in the spiral. - Omar E. Pol, Sep 10 2011
a(n), n >= 1, is also the inverse of 3, denoted by 3^(-1), Modd(2^(2*n - 1)). For Modd n see a comment on A203571. E.g., a(2) = 5, 3 * 5 = 15 == 1 (Modd 8), because floor(15/8) = 1 is odd and -15 == 1 (mod 8). For n = 1 note that 3 * 1 = 3 == 1 (Modd 2) because floor(3/2) = 1 and -3 == 1 (mod 2). The inverse of 3 taken Modd 2^(2*n) coincides with 3^(-1) (mod 2^(2*n)) given in A007583(n), n >= 1. - Wolfdieter Lang, Mar 12 2012
If an AVL tree has a leaf at depth n, then the tree can contain no more than a(n+1) nodes total. - Mike Rosulek, Nov 20 2012
Also, this is the Lucas sequence V(5, 4). - Bruno Berselli, Jan 10 2013
Also, for n > 0, a(n) is an odd number whose Collatz trajectory contains no odd number other than n and 1. - Jayanta Basu, Mar 24 2013
Sum_{n >= 1} 1/a(n) converges to (3*(log(4/3) - QPolyGamma[0, 1, 1/4]))/log(4) = 1.263293058100271... = A321873. - K. G. Stier, Jun 23 2014
Consider n spheres in R^n: the i-th one (i=1, ..., n) has radius r(i) = 2^(1-i) and the coordinates of its center are (0, 0, ..., 0, r(i), 0, ..., 0) where r(i) is in position i. The coordinates of the intersection point in the positive orthant of these spheres are (2/a(n), 4/a(n), 8/a(n), 16/a(n), ...). For example in R^2, circles centered at (1, 0) and (0, 1/2), and with radii 1 and 1/2, meet at (2/5, 4/5). - Jean M. Morales, May 19 2015
From Peter Bala, Oct 11 2015: (Start)
a(n) gives the values of m such that binomial(4*m + 1,m) is odd. Cf. A003714, A048716, A263132.
2*a(n) = A020988(n) gives the values of m such that binomial(4*m + 2, m) is odd.
4*a(n) = A080674(n) gives the values of m such that binomial(4*m + 4, m) is odd. (End)
Collatz Conjecture Corollary: Except for powers of 2, the Collatz iteration of any positive integer must eventually reach a(n) and hence terminate at 1. - Gregory L. Simay, May 09 2016
Number of active (ON, black) cells at stage 2^n - 1 of the two-dimensional cellular automaton defined by "Rule 598", based on the 5-celled von Neumann neighborhood. - Robert Price, May 16 2016
From Luca Mariot and Enrico Formenti, Sep 26 2016: (Start)
a(n) is also the number of coprime pairs of polynomials (f, g) over GF(2) where both f and g have degree n + 1 and nonzero constant term.
a(n) is also the number of pairs of one-dimensional binary cellular automata with linear and bipermutive local rule of neighborhood size n+1 giving rise to orthogonal Latin squares of order 2^m, where m is a multiple of n. (End)
Except for 0, 1 and 5, all terms are Brazilian repunits numbers in base 4, and so belong to A125134. For n >= 3, all these terms are composite because a(n) = {(2^n-1) * (2^n + 1)}/3 and either (2^n - 1) or (2^n + 1) is a multiple of 3. - Bernard Schott, Apr 29 2017
Given the 3 X 3 matrix A = [2, 1, 1; 1, 2, 1; 1, 1, 2] and the 3 X 3 unit matrix I_3, A^n = a(n)(A - I_3) + I_3. - Nicolas Patrois, Jul 05 2017
The binary expansion of a(n) (n >= 1) consists of n 1's alternating with n - 1 0's. Example: a(4) = 85 = 1010101_2. - Emeric Deutsch, Aug 30 2017
a(n) (n >= 1) is the viabin number of the integer partition [n, n - 1, n - 2, ..., 2, 1] (for the definition of viabin number see comment in A290253). Example: a(4) = 85 = 1010101_2; consequently, the southeast border of the Ferrers board of the corresponding integer partition is ENENENEN, where E = (1, 0), N = (0, 1); this leads to the integer partition [4, 3, 2, 1]. - Emeric Deutsch, Aug 30 2017
Numbers whose binary and Gray-code representations are both palindromes (i.e., intersection of A006995 and A281379). - Amiram Eldar, May 17 2021
Starting with n = 1 the sequence satisfies {a(n) mod 6} = repeat{1, 5, 3}. - Wolfdieter Lang, Jan 14 2022
Terms >= 5 are those q for which the multiplicative order of 2 mod q is floor(log_2(q)) + 2 (and which is 1 more than the smallest possible order for any q). - Tim Seuré, Mar 09 2024
The order of 2 modulo a(n) is 2*n for n >= 2. - Joerg Arndt, Mar 09 2024

			Apply Collatz iteration to 9: 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5 and hence 16, 8, 4, 2, 1.
Apply Collatz iteration to 27: 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5 and hence 16, 8, 4, 2, 1. [Corrected by _Sean A. Irvine_ at the suggestion of Stephen Cornelius, Mar 04 2024]
a(5) = (4^5 - 1)/3 = 341 = 11111_4 = {(2^5 - 1) * (2^5 + 1)}/3 = 31 * 33/3 = 31 * 11. - _Bernard Schott_, Apr 29 2017

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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).

T. D. Noe, Table of n, a(n) for n = 0..200
Peter Bala, A characterization of A002450, A020988 and A080674.
Henry Bottomley, Illustration of initial terms
Sung-Hyuk Cha, On Complete and Size Balanced k-ary Tree Integer Sequences, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75. - From _N. J. A. Sloane_, Dec 24 2012
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 8.
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
C. Ernst and D. W. Sumners, The Growth of the Number of Prime Knots, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987
Ernesto Estrada and José A. de la Peña, Integer sequences from walks in graphs, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 3, 78-84.
Rigoberto Flórez, Robinson A. Higuita, and Antara Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
Enrico Formenti and Luca Mariot, Exhaustive Generation of Linear Orthogonal CA, Automata 2023, Univ. Twente (Netherlands), see p. 22 of 38.
Mattia Fregola, Elementary Cellular Automata Rule 1 generating OEIS sequence A277799, A058896, A141725, A002450
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 373
Petro Kosobutskyy and Nataliia Nestor, On the mathematical model of the transformation of natural numbers by a function of a split type, Comp. Des. Sys. Theor. Practice (2024) Vol. 6, No. 2. See p. 7.
Petro Kosobutskyy, Anastasiia Yedyharova, and Taras Slobodzyan, From Newton's binomial and Pascal's triangle to Collatz's problem, Comp. Des. Sys., Theor. Practice (2023) Vol. 5, No. 1, 121-127.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
J. V. Leyendekkers and A.G. Shannon, Modular Rings and the Integer 3, Notes on Number Theory & Discrete Mathematics, 17 (2011), 47-51.
Luca Manzoni, Luca Mariot, and Giuliamaria Menara, Combinatorial Designs and Cellular Automata: A Survey, arXiv:2503.10320 [math.CO], 2025. See pp. 16, 27.
Luca Mariot, Cryptography by Cellular Automata, 2017.
Luca Mariot, Orthogonal labelings in de Bruijn graphs, IWOCA 2020 - Open Problems Session, Delft University of Technology (Netherlands).
Luca Mariot, Connections between Latin squares, Cellular Automata and Coprime Polynomials, Univ. Twente (Netherlands, 2023). See p. 16/37.
Luca Mariot and Enrico Formenti, The number of coprime/non-coprime pairs of polynomials over F_2 with degree n and nonzero constant term.
Luca Mariot, Enrico Formenti and Alberto Leporati, Constructing Orthogonal Latin Squares from Linear Cellular Automata. In: Exploratory papers of AUTOMATA 2016.
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Luca Mariot, Counting Coprime Polynomials over Finite Fields with Formal Languages and Compositions of Natural Numbers, Univ. Twente (Netherlands 2023). See p. 11.
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Index entries for linear recurrences with constant coefficients, signature (5,-4).

Partial sums of powers of 4, A000302.
When converted to binary, this gives A094028.
Subsequence of A003714.
Primitive factors: A129735.
Cf. A000225, A002446, A006995, A007583, A018215, A020988, A024036, A047849, A048716, A080355, A080674, A112627, A113860, A139391, A155701, A156605, A160967, A163834, A163868, A178415, A263132, A281379, A347834.

List([0..25], n -> (4^n-1)/3); # Muniru A Asiru, Feb 18 2018

a002450 = (`div` 3) . a024036
a002450_list = iterate ((+ 1) . (* 4)) 0
-- Reinhard Zumkeller, Oct 03 2012

[ (4^n-1)/3: n in [0..25] ]; // Klaus Brockhaus, Oct 28 2008

[n le 2 select n-1 else 5*Self(n-1)-4*Self(n-2): n in [1..70]]; // Vincenzo Librandi, Jun 13 2015

[seq((4^n-1)/3,n=0..40)];
A002450:=1/(4*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation, dropping the initial zero

Table[(4^n - 1)/3, {n, 0, 127}] (* Vladimir Joseph Stephan Orlovsky, Sep 29 2008 *)
LinearRecurrence[{5, -4}, {0, 1}, 30] (* Harvey P. Dale, Jun 23 2013 *)

makelist((4^n-1)/3, n, 0, 30); /* Martin Ettl, Nov 05 2012 */

PARI
```
a(n) = (4^n-1)/3;
```

my(z='z+O('z^40)); Vec(z/((1-z)*(1-4*z))) \\ Altug Alkan, Oct 11 2015

def A002450(n): return ((1<<(n<<1))-1)//3 # Chai Wah Wu, Jan 29 2023

((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)( * )).scanLeft(0: BigInt)( + ) // Alonso del Arte, Sep 17 2019

From Wolfdieter Lang, Apr 24 2001: (Start)
a(n+1) = Sum_{m = 0..n} A060921(n, m).
G.f.: x/((1-x)*(1-4*x)). (End)
a(n) = Sum_{k = 0..n-1} 4^k; a(n) = A001045(2*n). - Paul Barry, Mar 17 2003
E.g.f.: (exp(4*x) - exp(x))/3. - Paul Barry, Mar 28 2003
a(n) = (A007583(n) - 1)/2. - N. J. A. Sloane, May 16 2003
a(n) = A000975(2*n)/2. - N. J. A. Sloane, Sep 13 2003
a(n) = A084160(n)/2. - N. J. A. Sloane, Sep 13 2003
a(n+1) = 4*a(n) + 1, with a(0) = 0. - Philippe Deléham, Feb 25 2004
a(n) = Sum_{i = 0..n-1} C(2*n - 1 - i, i)*2^i. - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
a(n+1) = Sum_{k = 0..n} binomial(n+1, k+1)*3^k. - Paul Barry, Aug 20 2004
a(n) = center term in M^n * [1 0 0], where M is the 3 X 3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 0 0] = [A007583(n-1) a(n) A007583(n-1)]. E.g., a(4) = 85 since M^4 * [1 0 0] = [43 85 43] = [A007583(3) a(4) A007583(3)]. - Gary W. Adamson, Dec 18 2004
a(n) = Sum_{k = 0..n, j = 0..n} C(n, j)*C(j, k)*A001045(j - k). - Paul Barry, Feb 15 2005
a(n) = Sum_{k = 0..n} C(n, k)*A001045(n-k)*2^k = Sum_{k = 0..n} C(n, k)*A001045(k)*2^(n-k). - Paul Barry, Apr 22 2005
a(n) = A125118(n, 3) for n > 2. - Reinhard Zumkeller, Nov 21 2006
a(n) = Sum_{k = 0..n} 2^(n - k)*A128908(n, k), n >= 1. - Philippe Deléham, Oct 19 2008
a(n) = Sum_{k = 0..n} A106566(n, k)*A100335(k). - Philippe Deléham, Oct 30 2008
If we define f(m, j, x) = Sum_{k = j..m} binomial(m, k)*stirling2(k, j)*x^(m - k) then a(n-1) = f(2*n, 4, -2), n >= 2. - Milan Janjic, Apr 26 2009
a(n) = A014551(n) * A001045(n). - R. J. Mathar, Jul 08 2009
a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3) = 5*a(n-1) - 4*a(n-2), a(0) = 0, a(1) = 1, a(2) = 5. - Wolfdieter Lang, Oct 18 2010
a(0) = 0, a(n+1) = a(n) + 2^(2*n). - Washington Bomfim, Jan 21 2011
A036555(a(n)) = 2*n. - Reinhard Zumkeller, Jan 28 2011
a(n) = Sum_{k = 1..floor((n+2)/3)} C(2*n + 1, n + 2 - 3*k). - Mircea Merca, Jun 25 2011
a(n) = Sum_{i = 1..n} binomial(2*n + 1, 2*i)/3. - Wesley Ivan Hurt, Mar 14 2015
a(n+1) = 2^(2*n) + a(n), a(0) = 0. - Ben Paul Thurston, Dec 27 2015
a(k*n)/a(n) = 1 + 4^n + ... + 4^((k-1)*n). - Gregory L. Simay, Jun 09 2016
Dirichlet g.f.: (PolyLog(s, 4) - zeta(s))/3. - Ilya Gutkovskiy, Jun 26 2016
A000120(a(n)) = n. - André Dalwigk, Mar 26 2018
a(m) divides a(m*n), in particular: a(2*n) == 0 (mod 5), a(3*n) == 0 (mod 3*7), a(5*n) == 0 (mod 11*31), etc. - M. F. Hasler, Oct 19 2018
a(n) = 4^(n-1) + a(n-1). - Bob Selcoe, Jan 01 2020
a(n) = A178415(1, n) = A347834(1, n-1), arrays, for n >= 1.  - Wolfdieter Lang, Nov 29 2021
a(n) = A000225(2*n)/3. - John Keith, Jan 22 2022
a(n) = A080674(n) + 1 = A047849(n) - 1 = A163834(n) - 2 = A155701(n) - 3 = A163868(n) - 4 = A156605(n) - 7. - Ray Chandler, Jun 16 2023
From Peter Bala, Jul 23 2025: (Start)
The following are examples of telescoping products. Cf. A016153:
Product_{k = 1..2*n} 1 + 2^k/a(k+1) = a(n+1)/A007583(n) = (4^(n+1) - 1)/(2*4^n + 1).
Hence, Product_{k >= 1} 1 + 2^k/a(k+1) = 2.
Product_{k >= 1} 1 - 2^k/a(k+1) = 2/5, since 1 - 2^n/a(n+1) = b(n)/b(n-1), where b(n) = 2 - 3/(1 - 2^(n+1)).
Product_{k >= 1} 1 + (-2)^k/a(k+1) = 2/3, since 1 + (-2)^n/a(n+1) = c(n)/c(n-1), where c(n) = 2 - 1/(1 + (-2)^(n+1)).
Product_{k >= 1} 1 - (-2)^k/a(k+1) = 6/5, since 1 - (-2)^n/a(n+1) = d(n)/d(n-1), where d(n) = 2 - 1/(1 - (-2)^(n+1)). (End)

A007283 a(n) = 3*2^n.

Original entry on oeis.org

3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(3, 6), L(3, 6), P(3, 6), T(3, 6). See A008776 for definitions of Pisot sequences.
Numbers k such that A006530(A000010(k)) = A000010(A006530(k)) = 2. - Labos Elemer, May 07 2002
Also least number m such that 2^n is the smallest proper divisor of m which is also a suffix of m in binary representation, see A080940. - Reinhard Zumkeller, Feb 25 2003
Length of the period of the sequence Fibonacci(k) (mod 2^(n+1)). - Benoit Cloitre, Mar 12 2003
The sequence of first differences is this sequence itself. - Alexandre Wajnberg and Eric Angelini, Sep 07 2005
Subsequence of A122132. - Reinhard Zumkeller, Aug 21 2006
Apart from the first term, a subsequence of A124509. - Reinhard Zumkeller, Nov 04 2006
Total number of Latin n-dimensional hypercubes (Latin polyhedra) of order 3. - Kenji Ohkuma (k-ookuma(AT)ipa.go.jp), Jan 10 2007
Number of different ternary hypercubes of dimension n. - Edwin Soedarmadji (edwin(AT)systems.caltech.edu), Dec 10 2005
For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n + 1} -> {1, 2, 3} such that for fixed, different x_1, x_2,...,x_n in {1, 2, ..., n + 1} and fixed y_1, y_2,...,y_n in {1, 2, 3} we have f(x_i) <> y_i, (i = 1,2,...,n). - Milan Janjic, May 10 2007
a(n) written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, n times 0 (see A003953). - Jaroslav Krizek, Aug 17 2009
Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010
Numbers containing the number 3 in their Collatz trajectories. - Reinhard Zumkeller, Feb 20 2012
a(n-1) gives the number of ternary numbers with n digits with no two adjacent digits in common; e.g., for n=3 we have 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210 and 212. - Jon Perry, Oct 10 2012
If n > 1, then a(n) is a solution for the equation sigma(x) + phi(x) = 3x-4. This equation also has solutions 84, 3348, 1450092, ...  which are not of the form 3*2^n. - Farideh Firoozbakht, Nov 30 2013
a(n) is the upper bound for the "X-ray number" of any convex body in E^(n + 2), conjectured by Bezdek and Zamfirescu, and proved in the plane E^2 (see the paper by Bezdek and Zamfirescu). - L. Edson Jeffery, Jan 11 2014
If T is a topology on a set V of size n and T is not the discrete topology, then T has at most 3 * 2^(n-2) many open sets. See Brown and Stephen references. - Ross La Haye, Jan 19 2014
Comment from Charles Fefferman, courtesy of Doron Zeilberger, Dec 02 2014: (Start)
Fix a dimension n. For a real-valued function f defined on a finite set E in R^n, let Norm(f, E) denote the inf of the C^2 norms of all functions F on R^n that agree with f on E. Then there exist constants k and C depending only on the dimension n such that Norm(f, E) <= C*max{ Norm(f, S) }, where the max is taken over all k-point subsets S in E.  Moreover, the best possible k is 3 * 2^(n-1).
The analogous result, with the same k, holds when the C^2 norm is replaced, e.g., by the C^1, alpha norm (0 < alpha <= 1). However, the optimal analogous k, e.g., for the C^3 norm is unknown.
For the above results, see Y. Brudnyi and P. Shvartsman (1994). (End)
Also, coordination sequence for (infinity, infinity, infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
The average of consecutive powers of 2 beginning with 2^1. - Melvin Peralta and Miriam Ong Ante, May 14 2016
For n > 1, a(n) is the smallest Zumkeller number with n divisors that are also Zumkeller numbers (A083207). - Ivan N. Ianakiev, Dec 09 2016
Also, for n >= 2, the number of length-n strings over the alphabet {0,1,2,3} having only the single letters as nonempty palindromic subwords. (Corollary 21 in Fleischer and Shallit) - Jeffrey Shallit, Dec 02 2019
Also, a(n) is the minimum link-length of any covering trail, circuit, path, and cycle for the set of the 2^(n+2) vertices of an (n+2)-dimensional hypercube. - Marco Ripà, Aug 22 2022
The known fixed points of maps n -> A163511(n) and n -> A243071(n). [See comments in A163511]. - Antti Karttunen, Sep 06 2023
The finite subsequence a(3), a(4), a(5), a(6) = 24, 48, 96, 192 is one of only two geometric sequences that can be formed with all interior angles (all integer, in degrees) of a simple polygon. The other sequence is a subsequence of A000244 (see comment there). - Felix Huber, Feb 15 2024
A level 1 Sierpiński triangle is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles.  For n>2, a(n-3) is the radius of the level n Sierpiński triangle graph. - Allan Bickle, Aug 03 2024

Jason I. Brown, Discrete Structures and Their Interactions, CRC Press, 2013, p. 71.
T. Ito, Method, equipment, program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (written in Japanese, a(2)=12, a(3)=24, a(4)=48, a(5)=96, a(6)=192, a(7)=384 (a(7)=284 was corrected)).
Kenji Ohkuma, Atsuhiro Yamagishi and Toru Ito, Cryptography Research Group Technical report, IT Security Center, Information-Technology Promotion Agency, JAPAN.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. Bezdek and Tudor Zamfirescu, A Characterization of 3-dimensional Convex Sets with an Infinite X-ray Number, in: Coll. Math. Soc. J. Bolyai 63., Intuitive Geometry, Szeged (Hungary), North-Holland, Amsterdam, 1991, pp. 33-38.
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney's extension theorem, International Mathematics Research Notices 1994.3 (1994): 129-139.
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Tomislav Došlić, Kepler-Bouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.
John Elias, Illustration: 2^n+1 hexagram perimeters
Lukas Fleischer and Jeffrey Shallit, Words With Few Palindromes, Revisited, arxiv preprint arXiv:1911.12464 [cs.FL], November 27 2019.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Tanya Khovanova, Recursive Sequences
Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
Edwin Soedarmadji, Latin Hypercubes and MDS Codes, Discrete Mathematics, Volume 306, Issue 12, Jun 28 2006, Pages 1232-1239
D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739 - 741, 1968.
Index entries for linear recurrences with constant coefficients, signature (2).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
Subsequence of the following sequences: A029744, A029747, A029748, A029750, A362804 (after 3), A364494, A364496, A364289, A364291, A364292, A364295, A364497, A364964, A365422.
Essentially same as A003945 and A042950.
Row sums of (5, 1)-Pascal triangle A093562 and of (1, 5) Pascal triangle A096940.
Cf. A000079, A100540, A124508, A221718.
Cf. Latin squares: A000315, A002860, A003090, A040082, A003191; Latin cubes: A098843, A098846, A098679, A099321.
Cf. A133466, A225546, A163511, A243071.
Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959.

a007283 = (* 3) . (2 ^)
a007283_list = iterate (* 2) 3
-- Reinhard Zumkeller, Mar 18 2012, Feb 20 2012

[3*2^n: n in [0..30]]; // Vincenzo Librandi, May 18 2011

A007283:=n->3*2^n; seq(A007283(n), n=0..50); # Wesley Ivan Hurt, Dec 03 2013

Table[3(2^n), {n, 0, 32}] (* Alonso del Arte, Mar 24 2011 *)

A007283(n):=3*2^n$
makelist(A007283(n),n,0,30); /* Martin Ettl, Nov 11 2012 */

PARI
```
a(n)=3*2^n
```

a(n)=3<Charles R Greathouse IV, Oct 10 2012

def A007283(n): return 3<Chai Wah Wu, Feb 14 2023

(List.fill(40)(2: BigInt)).scanLeft(1: BigInt)( * ).map(3 * ) // _Alonso del Arte, Nov 28 2019

G.f.: 3/(1-2*x).
a(n) = 2*a(n - 1), n > 0; a(0) = 3.
a(n) = Sum_{k = 0..n} (-1)^(k reduced (mod 3))*binomial(n, k). - Benoit Cloitre, Aug 20 2002
a(n) = A118416(n + 1, 2) for n > 1. - Reinhard Zumkeller, Apr 27 2006
a(n) = A000079(n) + A000079(n + 1). - Zerinvary Lajos, May 12 2007
a(n) = A000079(n)*3. - Omar E. Pol, Dec 16 2008
From Paul Curtz, Feb 05 2009: (Start)
a(n) = b(n) + b(n+3) for b = A001045, A078008, A154879.
a(n) = abs(b(n) - b(n+3)) with b(n) = (-1)^n*A084247(n). (End)
a(n) = 2^n + 2^(n + 1). - Jaroslav Krizek, Aug 17 2009
a(n) = A173786(n + 1, n) = A173787(n + 2, n). - Reinhard Zumkeller, Feb 28 2010
A216022(a(n)) = 6 and A216059(a(n)) = 7, for n > 0. - Reinhard Zumkeller, Sep 01 2012
a(n) = (A000225(n) + 1)*3. - Martin Ettl, Nov 11 2012
E.g.f.: 3*exp(2*x). - Ilya Gutkovskiy, May 15 2016
A020651(a(n)) = 2. - Yosu Yurramendi, Jun 01 2016
a(n) = sqrt(A014551(n + 1)*A014551(n + 2) + A014551(n)^2). - Ezhilarasu Velayutham, Sep 01 2019
a(A048672(n)) = A225546(A133466(n)). - Michel Marcus and Peter Munn, Nov 29 2019
Sum_{n>=1} 1/a(n) = 2/3. - Amiram Eldar, Oct 28 2020

The sequence (-1)^n*a(n) is the inverse binomial transform of A166956.

The main diagonal of the table of a(n) and its higher differences in successive rows is 0,4,8,16,32,.. , 4*A131577(n).

The sequence is a Lucas sequence V(P,Q) with P = 5 and Q = 4, so if n is a prime number, then V_n(5,4) - 5 is divisible by n. The smallest pseudoprime q which divides V_q(5,4) - 5 is 15.
Also the edge cover number of the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Sep 20 2017
First bisection of A000051, A049332, A052531 and A014551. - Klaus Purath, Sep 23 2020

Sum of divisors of 4^n. - Paul Barry, Oct 13 2005
Subsequence of A000069; A132680(a(n)) = A005408(n). - Reinhard Zumkeller, Aug 26 2007
If x = a(n), y = A000079(n+1) and z = A087289(n), then x^2 + 2*y^2 = z^2. - Vincenzo Librandi, Jun 09 2014
It seems that a(n) divides A001676(3+4n). Several other entries apparently have this sequence embedded in them, e.g., A014551, A168604, A213243, A213246-8, and A279872. - Tom Copeland, Dec 27 2016
To elaborate on Librandi's comment from 2014: all these numbers, even if prime in Z, are sure not to be prime in Z[sqrt(2)], since a(n) can at least be factored as ((2^(2n + 1) - 1) - (2^(2n) - 1)*sqrt(2))((2^(2n + 1) - 1) + (2^(2n) - 1)*sqrt(2)). For example, 7 = (3 - sqrt(2))(3 + sqrt(2)), 31 = (7 - 3*sqrt(2))(7 + 3*sqrt(2)), 127 = (15 - 7*sqrt(2))(15 + 7*sqrt(2)). - Alonso del Arte, Oct 17 2017
Largest odd factors of A147590. - César Aguilera, Jan 07 2020

The identity 2 = 2^2/3 + 2^3/(3*3) - 2^4/(3*3*9) - 2^5/(3*3*9*15) + + - - can be viewed as a generalized Engel-type expansion of the number 2 to the base 2. Compare with A014551. - Peter Bala, Nov 13 2013

Number of positive integers requiring exactly n signed bits in the modified non-adjacent form representation. - Ralf Stephan, Aug 02 2003
The n-th entry (n>1) of the sequence is equal to the 1,1-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
Pisano period lengths:  1, 1, 6, 2, 2, 6, 6, 2, 18, 2, 10, 6, 12, 6, 6, 2, 8, 18, 18, 2, ... - R. J. Mathar, Aug 10 2012
For n >= 1, a(n) is the number of ways to tile a strip of length n+2 with blue squares and blue and red dominos, with the restriction that the first two tiles must be the same color. - Guanji Chen and Greg Dresden, Jul 15 2024

4*2^n = A002249(n)^2 + 7*A001607(n)^2. See A077020, A077021.
Among presented initial elements of the sequence a(n), the maximal increasing or decreasing subsequences have length either 3 or 4. - Roman Witula, Aug 21 2012
This is the Lucas Sequence V_n(P, Q) = V_n(1, 2). U_n(1, 2) = A107920(n). - Raphie Frank, Dec 25 2013
The only numbers that occur more than once are 1=a(1)=a(4) and -5=a(3)=a(9).  See Noam D. Elkies's posting in the Mathematics Stack Exchange link. - Robert Israel, Dec 21 2016

cp's OEIS Frontend

A166977 Jacobsthal-Lucas numbers A014551, except a(0) = 0.

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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.

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A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

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A002450 a(n) = (4^n - 1)/3.

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A007283 a(n) = 3*2^n.

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