This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A001045 M2482 N0983 #1268 Aug 25 2025 02:15:58
%S A001045 0,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691,
%T A001045 87381,174763,349525,699051,1398101,2796203,5592405,11184811,22369621,
%U A001045 44739243,89478485,178956971,357913941,715827883,1431655765,2863311531,5726623061,11453246123
%N 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.
%C A001045 _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
%C A001045 Number of ways to tile a 3 X (n-1) rectangle with 1 X 1 and 2 X 2 square tiles.
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 Arises in study of sorting by merge insertions and in analysis of a method for computing GCDs - see Knuth reference.
%C A001045 Number of perfect matchings of a 2 X n grid upon replacing unit squares with tetrahedra (C_4 to K_4):
%C A001045 o----o----o----o...
%C A001045 | \/ | \/ | \/ |
%C A001045 | /\ | /\ | /\ |
%C A001045 o----o----o----o... - _Roberto E. Martinez II_, Jan 07 2002
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 Sums of pairs of consecutive terms give all powers of 2 in increasing order. - _Amarnath Murthy_, Aug 15 2002
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 Number of positive integers requiring exactly n signed bits in the nonadjacent form representation.
%C A001045 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
%C A001045 Counts walks between adjacent vertices of a triangle. - _Paul Barry_, Nov 17 2003
%C A001045 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
%C A001045 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
%C A001045 Number of permutations with no fixed points avoiding 231 and 132.
%C A001045 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
%C A001045 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
%C A001045 a(n+1) gives row sums of A059260. - _Paul Barry_, Jan 26 2005
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 A147612(a(n)) = 1. - _Reinhard Zumkeller_, Nov 08 2008
%C A001045 a(n+1) = Sum(A153778(i): 2^n <= i < 2^(n+1)). - _Reinhard Zumkeller_, Jan 01 2009
%C A001045 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
%C A001045 Number of pairs of consecutive odious (or evil) numbers between 2^(n+1) and 2^(n+2), inclusive. - _T. D. Noe_, Feb 05 2009
%C A001045 Equals eigensequence of triangle A156319. - _Gary W. Adamson_, Feb 07 2009
%C A001045 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
%C A001045 Starting with offset 1 = INVERTi transform of A002605: (1, 2, 6, 16, 44, ...). - _Gary W. Adamson_, May 12 2009
%C A001045 Convolved with (1, 2, 2, 2, ...) = A000225: (1, 3, 7, 15, 31, ...). - _Gary W. Adamson_, May 23 2009
%C A001045 The product of a pair of successive terms is always a triangular number. - _Giuseppe Ottonello_, Jun 14 2009
%C A001045 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
%C A001045 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
%C A001045 As a fraction: 1/88 = 0.0113636363... or 1/9898 = 0.00010103051121... - _Mark Dols_, May 18 2010
%C A001045 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
%C A001045 Rule 28 elementary cellular automaton (A266508) generates this sequence. - _Paul Muljadi_, Jan 27 2011
%C A001045 This is a divisibility sequence. - _Michael Somos_, Feb 06 2011
%C A001045 From _L. Edson Jeffery_, Apr 04 2011: (Start)
%C A001045 Let U be the unit-primitive matrix (see [Jeffery])
%C A001045 U = U_(6,2) =
%C A001045 (0 0 1)
%C A001045 (0 2 0)
%C A001045 (2 0 1).
%C A001045 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)
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 From _Paul Curtz_, Dec 11 2012: (Start)
%C A001045 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, ...
%C A001045 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:
%C A001045 0 1/2 1/2 3/2 5/2 11/2 ...
%C A001045 1/2 0 1 1 3 5 ...
%C A001045 -1/2 1 0 2 2 6 ...
%C A001045 3/2 -1 2 0 4 4 ...
%C A001045 -5/2 3 -2 4 0 8 ...
%C A001045 11/2 -5 6 -4 8 0 ...
%C A001045 The main diagonal in this array contains 0's. (End)
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 Number of conjugacy classes of Z/2Z X Z/2Z in GL(2,2^(n+1)). - _Jared Warner_, Aug 18 2013
%C A001045 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
%C A001045 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
%C A001045 This is the Lucas sequence U(1, -2). - _Felix P. Muga II_, Mar 21 2014
%C A001045 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
%C A001045 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
%C A001045 From _César Eliud Lozada_, Jan 21 2015: (Start)
%C A001045 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].
%C A001045 Then, for n >= 1, Complement(Complement(...(Complement(P))..)) = (n times) =
%C A001045 [2*a(n-1)*a*x + (2*a(n-1) - (-1)^n)*(b*y + c*z):
%C A001045 2*a(n-1)*b*y + (2*a(n-1) - (-1)^n)*(c*z + a*x):
%C A001045 2*a(n-1)*c*z + (2*a(n-1) - (-1)^n)*(a*x + b*y)]. (End)
%C A001045 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
%C A001045 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
%C A001045 With the quantum integers defined by [n+1]_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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 Also the number of independent vertex sets and vertex covers in the 2 X (n-2) king graph. - _Eric W. Weisstein_, Sep 21 2017
%C A001045 From _César Eliud Lozada_, Dec 14 2017: (Start)
%C A001045 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.
%C A001045 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)
%C A001045 a(n) is also the number of derangements in S_{n+1} with empty peak set. - _Isabella Huang_, Apr 01 2018
%C A001045 For n > 0, gcd(a(n), a(n+1)) = 1. - _Kengbo Lu_, Jul 27 2020
%C A001045 Number of 2-compositions of n+1 with 1 not allowed as a part; see Hopkins & Ouvry reference. - _Brian Hopkins_, Aug 17 2020
%C A001045 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
%C A001045 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
%C A001045 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
%C A001045 Named by Horadam (1988) after the German mathematician Ernst Jacobsthal (1882-1965). - _Amiram Eldar_, Oct 02 2023
%C A001045 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
%C A001045 From _Miquel A. Fiol_, May 25 2024: (Start)
%C A001045 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).
%C A001045 For instance, for n=3, the a(3)=3 states (3-coloring of the terminals) of C_4 are
%C A001045 a a a a a b
%C A001045 a a b b a b (End)
%C A001045 Also, with offset 1, the cogrowth sequence of the 6-element dihedral group D3. - _Sean A. Irvine_, Nov 04 2024
%D A001045 Jathan Austin and Lisa Schneider, Generalized Fibonacci sequences in Pythagorean triple preserving sequences, Fib. Q., 58:1 (2020), 340-350.
%D A001045 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.
%D A001045 International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest Problem #16.
%D A001045 Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World Scientific Press, 2007. See p. 80.
%D A001045 Ernst Erich Jacobsthal, Fibonaccische Polynome und Kreisteilungsgleichungen, Sitzungsber. Berliner Math. Gesell. 17 (1919-1920), 43-57.
%D A001045 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.
%D A001045 Donald E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.3.1, Eq. 13.
%D A001045 Thomas Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 98.
%D A001045 Steven Roman, Introduction to Coding and Information Theory, Springer Verlag, 1996, 41-42.
%D A001045 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]
%D A001045 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001045 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001045 Robert M. Young, Excursions in Calculus, MAA, 1992, p. 239
%H A001045 Indranil Ghosh, <a href="/A001045/b001045.txt">Table of n, a(n) for n = 0..3316</a> (terms 0..500 from T. D. Noe)
%H A001045 M. H. Albert, M. D. Atkinson, and V. Vatter, <a href="https://arxiv.org/abs/1209.0425">Inflations of geometric grid classes: three case studies</a>.
%H A001045 Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, and Jiemeng Zhang, <a href="https://arxiv.org/abs/1911.01687">Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences</a>, arXiv:1911.01687 [math.CO], 2019.
%H A001045 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp.315-318.
%H A001045 Mohammad K. Azarian, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/28-2/elementary28-2.pdf">Limit of Nested Square Roots, Problem B-664</a>, Fibonacci Quarterly, Vol. 28, No. 2, May 1990, p. 182.
%H A001045 Mohammad K. Azarian, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/29-2/elementary29-2.pdf">Solution to Problem B-664, Limit of Nested Square Roots</a>, Fibonacci Quarterly, Vol. 29, No. 2, May 1991, p. 182.
%H A001045 Roland Bacher, <a href="https://arxiv.org/abs/1509.09054">Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle</a>, arXiv:1509.09054 [math.CO], 2015.
%H A001045 Paul Barry, <a href="https://arxiv.org/abs/1205.2565">On sequences with {-1, 0, 1} Hankel transforms</a>, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012
%H A001045 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A001045 Paul Barry, <a href="https://www.cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%H A001045 Paul Barry, <a href="https://doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%H A001045 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19 (2016), Article 16.3.5.
%H A001045 Paul Barry, <a href="https://arxiv.org/abs/2504.09719">Notes on Riordan arrays and lattice paths</a>, arXiv:2504.09719 [math.CO], 2025. See pp. 14, 29.
%H A001045 Katerina Böhmová, Cristina Dalfó, and Clemens Huemer, <a href="https://arxiv.org/abs/1512.05917">On cyclic Kautz digraphs</a>, arXiv:1512.05917 [math.CO], 2015; <a href="http://upcommons.upc.edu/bitstream/handle/2117/80848/Kautz-subdigraphs.pdf">alternative link</a>.
%H A001045 Wieb Bosma, <a href="https://doi.org/10.5802/jtnb.301">Signed bits and fast exponentiation</a>, J. Th. Nombres de Bordeaux, 13 no. 1 (2001), p. 27-41.
%H A001045 Garry Bowlin and Matthew G. Brin, <a href="https://arxiv.org/abs/1301.3984">Coloring Planar Graphs via Colored Paths in the Associahedra</a>, arXiv preprint arXiv:1301.3984 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 12 2013
%H A001045 Rachael Boyd and Richard Hepworth, <a href="https://arxiv.org/abs/2006.04261">Combinatorics of injective words for Temperley-Lieb algebras</a>, arXiv:2006.04261 [math.AT], 2020.
%H A001045 Paul Bradley and Peter Rowley, <a href="http://eprints.ma.man.ac.uk/2167/01/covered/MIMS_ep2014_42.pdf">Orbits on k-subsets of 2-transitive Simple Lie-type Groups</a>, 2014.
%H A001045 Henri Brocard, <a href="https://archive.org/details/nouvellecorresp00mansgoog/page/145/mode/1up?view=theater">Propriété d'une série de triangles</a>, Nouvelle Correspondance Mathématique, Vol. 6 (1880), 145-151.
%H A001045 H. Bruhn, L. Gellert, and J. Günther, <a href="https://arxiv.org/abs/1503.03390">Jacobsthal numbers in generalised Petersen graphs</a>, arXiv preprint arXiv:1503.03390 [math.CO], 2015.
%H A001045 H. Bruhn, L. Gellert, and J. Günther, <a href="http://eurocomb2015.b.uib.no/files/2015/08/endm1943.pdf">Jacobsthal numbers in generalised Petersen graphs</a>, Electronic Notes in Discrete Math., 2015.
%H A001045 Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Falcao/falcao5.html">Intrinsic Properties of a Non-Symmetric Number Triangle</a>, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
%H A001045 L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-5/carlitz3-a.pdf">Representations for a special sequence</a>, Fibonacci Quarterly 10.5 (1972), pages 499-518 and 550.
%H A001045 Paula Catarino, Helena Campos, and Paulo Vasco. <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_46_from37to53.pdf">On the Mersenne sequence</a>. Annales Mathematicae et Informaticae, 46 (2016) pp. 37-53.
%H A001045 Miquel Cerda, <a href="/A001045/a001045.jpg">Irregular triangle whose row sums are the Jacobsthal numbers</a>.
%H A001045 Ji Young Choi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Choi/choi6.html">Ternary Modified Collatz Sequences And Jacobsthal Numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.7.5.
%H A001045 Ji Young Choi, <a href="https://www.emis.de/journals/JIS/VOL21/Choi/choi10.html">A Generalization of Collatz Functions and Jacobsthal Numbers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
%H A001045 Johann Cigler, <a href="https://arxiv.org/abs/1501.04750">Some remarks and conjectures related to lattice paths in strips along the x-axis</a>, arXiv:1501.04750 [math.CO], 2015.
%H A001045 M. H. Cilasun, <a href="https://arxiv.org/abs/1412.3265">An Analytical Approach to Exponent-Restricted Multiple Counting Sequences</a>, arXiv preprint arXiv:1412.3265 [math.NT], 2014.
%H A001045 M. H. Cilasun, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Cilasun/cila5.html">Generalized Multiple Counting Jacobsthal Sequences of Fermat Pseudoprimes</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.2.3.
%H A001045 Charles K. Cook and Michael R. Bacon, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from27to39.pdf">Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 27-39.
%H A001045 D. E. Daykin, D. J. Kleitman, and D. B. West, <a href="https://doi.org/10.1016/0097-3165(79)90063-3">The number of meets between two subsets of a lattice</a>, J. Combin. Theory, A 26 (1979), 135-156.
%H A001045 Gesualdo Delfino and Jacopo Viti, <a href="https://arxiv.org/abs/1104.4323">Potts q-color field theory and scaling random cluster model</a>, arXiv:1104.4323 [hep-th], 2011.
%H A001045 Karl Dilcher and Larry Ericksen, <a href="https://doi.org/10.1007/s11139-016-9864-3">Continued fractions and Stern polynomials</a>. Ramanujan Journal (2017). See Table 2.
%H A001045 Karl Dilcher and Hayley Tomkins, <a href="http://math.colgate.edu/~integers/s29/s29.mail.html">Square classes and divisibility properties of Stern polynomials</a>, Integers (2018) 18, Article #A29.
%H A001045 Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="https://arxiv.org/abs/1503.04249">Odd-Rule Cellular Automata on the Square Grid</a>, arXiv:1503.04249 [math.CO], 2015.
%H A001045 Peter E. Finch, <a href="https://arxiv.org/abs/1201.4470">From spin to anyon notation: The XXZ Heisenberg model as a D_3 (or su(2)_4) anyon chain</a>, arXiv preprint arXiv:1201.4470 [math-ph], 2012.
%H A001045 M. A. Fiol and J. Vilaltella, <a href="https://doi.org/10.37236/3629">Some results on the structure of multipoles in the study of snarks</a>, Electron J. Combin., 22(1) (2015), #P1.45.
%H A001045 M. Cetin Firengiz and A. Dil, <a href="http://www.nntdm.net/papers/nntdm-20/NNTDM-20-4-21-32.pdf">Generalized Euler-Seidel method for second order recurrence relations</a>, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
%H A001045 Darrin D. Frey and James A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/SELLERS/sellers.html">Jacobsthal Numbers and Alternating Sign Matrices</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.2.3.
%H A001045 Robert Frontczak and Taras Goy, <a href="https://doi.org/10.15330/cmp.12.1.34-45">Mersenne-Horadam identities using generating functions</a>, Carpathian Math. Publ. (2020) Vol. 12, No. 1, 34-45.
%H A001045 E. M. García-Caballero, S. G. Moreno, and M. P. Prophet, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-1/CaballeroMorenoProphet.pdf">New Viète-like infinite products of nested radicals</a>, Fib. Q., 52 (No. 1, 2014), 27-31.
%H A001045 Dale Gerdemann, <a href="https://www.youtube.com/watch?v=9fAHnqeAFlQ">Triangles from (1,2) Recursion</a> and <a href="https://www.youtube.com/watch?v=yCz78MnALxM">Recursive Triangle Fragmentation from (1,2) Recursion </a>, YouTube Videos, December 2014.
%H A001045 A. Goldberg and N. A. Rosenberg, <a href="https://doi.org/10.1534/genetics.115.178509">Beyond 2/3 and 1/3: the complex signatures of sex-biased admixture on the X chromosome</a>, Genetics 201 (2015), 263-279.
%H A001045 J. Goldwasser et al., <a href="https://doi.org/10.1016/S0012-365X(98)00373-2">The density of ones in Pascal's rhombus</a>, Discrete Math., 204 (1999), 231-236.
%H A001045 Taras Goy, <a href="https://doi.org/10.32523/2077-9879-2018-9-4-61-67">On determinants and permanents of some Toeplitz-Hessenberg matrices whose entries are Jacobsthal numbers</a>, Eurasian Math. J. (2018) Vol. 9, No. 4, 61-67.
%H A001045 Taras P. Goy, <a href="http://lib.pu.if.ua:8080/bitstream/123456789/2610/1/MSTU8-pages.pdf">Jacobsthal number identities using the generalized Brioschi formula</a>, Vasyl Stefanyk Precarpathian National University, (Ivano-Frankivsk, Ukraine, 2020).
%H A001045 Jonny Griffiths, <a href="https://doi.org/10.1017/mag.2015.51">The Not-on-the-line transformation</a>, The Mathematical Gazette, 99, pp 347-352 (2015).
%H A001045 Martin Griffiths and Alex Bramham, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/53-2/GriffithsBramham11172014.pdf">The Jacobsthal Numbers: Two Results and Two Questions</a>, Fibonacci Quart. 53 (2015), no. 2, 147-151.
%H A001045 Richard K. Guy, <a href="/A002186/a002186.pdf">Letters to N. J. A. Sloane, June-August 1968</a>.
%H A001045 Richard K. Guy, Tanya Khovanova, and Julian Salazar, <a href="https://arxiv.org/abs/1207.5099">Conway's subprime Fibonacci sequences</a>, arXiv:1207.5099 [math.NT], 2012-2014.
%H A001045 Lee Hae-hwang, <a href="/A026644/a026644.html">Illustration of initial terms in terms of rosemary plants</a>.
%H A001045 Silvia Heubach, <a href="http://www.calstatela.edu/sites/default/files/users/u1231/Papers/cgtc30.pdf">Tiling an m-by-n area with squares of size up to k-by-k (m<=5)</a>, preprint, published in: Congressus Numerantium 140 (1999), 43-64.
%H A001045 A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, <a href="https://doi.org/10.1007/978-3-0348-0237-6">The Tower of Hanoi - Myths and Maths</a>, Birkhäuser 2013. See page 56. <a href="http://tohbook.info">Book's website</a>.
%H A001045 Andreas M. Hinz and Paul K. Stockmeyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Hinz/hinz5.html">Precious Metal Sequences and Sierpinski-Type Graphs</a>, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
%H A001045 Brian Hopkins and Stéphane Ouvry, <a href="https://arxiv.org/abs/2008.04937">Combinatorics of Multicompositions</a>, arXiv:2008.04937 [math.CO], 2020.
%H A001045 Brian Hopkins, Mark Shattuck, Andrew V. Sills, Thotsaporn "AEK" Thanatipanonda, and Hua Wang, <a href="https://thotsaporn.com/composition.pdf">Parts and subword patterns in compositions</a>, Preprint 2015.
%H A001045 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/26-1/horadam2.pdf">Jacobsthal and Pell Curves</a>, Fib. Quart. 26, 79-83, 1988. [See J_n.]
%H A001045 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/34-1/horadam2.pdf">Jacobsthal Representation Numbers</a>, Fib Quart. 34, 40-54, 1996.
%H A001045 Jia Huang and Erkko Lehtonen, <a href="https://arxiv.org/abs/2401.15786">Associative-commutative spectra for some varieties of groupoids</a>, arXiv:2401.15786 [math.CO], 2024. See p. 17.
%H A001045 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=142">Encyclopedia of Combinatorial Structures 142</a>.
%H A001045 Dan Ismailescu, Joehyun Kim, Kelvin Kim, and Jeewoo Lee, <a href="https://arxiv.org/abs/1908.02749">The largest angle bisection procedure</a>, arXiv:1908.02749 [math.MG], 2019. See p. 17.
%H A001045 Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, J. Int. Seq. 18 (2015), Article 15.4.7.
%H A001045 Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), Article 18.1.4.
%H A001045 L. Edson Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>.
%H A001045 D. Jhala, G. P. S. Rathore, and K. Sisodiya, <a href="https://doi.org/10.12691/tjant-2-4-3">Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes</a>, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124.
%H A001045 Tanya Khovanova, <a href="https://arxiv.org/abs/1801.01143">Coins and Logic</a>, arXiv:1801.01143 [math.HO], 2018.
%H A001045 Tanya Khovanova and Konstantin Knop, <a href="https://arxiv.org/abs/1611.09201">Coins that Change Their Weights</a>, arXiv:1611.09201 [math.CO], 2016.
%H A001045 Sandi Klavžar, <a href="https://doi.org/10.1007/s10878-011-9433-z">Structure of Fibonacci cubes: a survey</a>, J. Comb. Optim., 25, 2013, 505-522.
%H A001045 Masato Kobayashi, <a href="https://arxiv.org/abs/1907.11802">Weighted counting of Bruhat paths by shifted R-polynomials</a>, arXiv:1907.11802 [math.CO], 2019.
%H A001045 Takao Komatsu, <a href="https://arxiv.org/abs/1903.09986">Continued fractions associated with the topological index of the caterpillar-bond graph</a>, arXiv:1903.09986 [math.CO], 2019.
%H A001045 Thomas Koshy and Ralph P. Grimaldi, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/55-2/KoshyGrimaldi10272016.pdf">Ternary words and Jacobsthal numbers</a>, Fib. Quart., 55 (No. 2, 2017), 129-136.
%H A001045 Peter J. Larcombe, Julius Fergy T. Rabago, and Eric J. Fennessey, <a href="http://pjm.ppu.edu/sites/default/files/papers/PJM_April2018_397to405.pdf">On two derivative sequences from scaled geometric mean sequence terms</a>, Palestine Journal of Mathematics (2018) Vol. 7(2), 397-405.
%H A001045 Kyu-Hwan Lee and Se-jin Oh, <a href="https://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%H A001045 E. Lemmen, J. van Duivenbode, J. L. Duarte, and E. A. Lomonova, <a href="https://doi.org/10.1109/JESTPE.2015.2427913">Flexible Multilevel Converters using 4-Switch Extended Commutation Cells</a>, Emerging and Selected Topics in Power Electronics, IEEE Journal of, Volume:PP, Issue: 99, 2015.
%H A001045 S. L. Levine, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/26-4/levine.pdf">Suppose more rabbits are born</a>, Fib. Quart., 26 (1988), 306-311.
%H A001045 Alan J. Macfarlane, <a href="http://www.damtp.cam.ac.uk/user/ajm/Papers2016/CellularAutomatonRule150.ps">On generating functions of some sequences of integers defined in the evolution of the cellular automaton Rule 150</a>, Preprint 2016.
%H A001045 T. Mansour and A. Robertson, <a href="https://arxiv.org/abs/math/0204005">Refined restricted permutations avoiding subsets of patterns of length three</a>, arXiv:math/0204005 [math.CO], 2002.
%H A001045 B. D. McKay and I. M. Wanless, <a href="https://doi.org/10.1137/070693874">A census of small latin hypercubes</a>, SIAM J. Discrete Math. 22, (2008) 719-736.
%H A001045 A. Moghaddamfar and H. Tajbakhsh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Moghaddamfar/moghadam3.html">More Determinant Representations for Sequences</a>, Journal of Integer Sequences, 17 (2014), #14.5.6.
%H A001045 Sophie Morier-Genoud, <a href="https://arxiv.org/abs/1907.12790">Counting Coxeter's friezes over a finite field via moduli spaces</a>, arXiv:1907.12790 [math.CO], 2019.
%H A001045 F. P. Muga II, <a href="https://www.researchgate.net/publication/267327689">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, March 2014; Preprint on ResearchGate.
%H A001045 Emanuele Munarini, <a href="https://doi.org/10.1515/puma-2015-0028">A generalization of André-Jeannin's symmetric identity</a>, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 98-118.
%H A001045 G. Myerson and A. J. van der Poorten, <a href="https://www.jstor.org/stable/2974639">Some problems concerning recurrence sequences</a>, Amer. Math. Monthly 102 (1995), no. 8, 698-705.
%H A001045 S. Northshield, <a href="https://doi.org/10.4169/000298910X496714">Stern's diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ...</a>, Amer. Math. Monthly, 117 (2010), 581-598.
%H A001045 Kritkhajohn Onphaeng and Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Pongsriiam/pong11.html">Jacobsthal and Jacobsthal-Lucas Numbers and Sums Introduced by Jacobsthal and Tverberg</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.6.
%H A001045 Ahmet Öteleş, Zekeriya Y. Karatas, and Diyar O. Mustafa Zangana, <a href="https://www.emis.de/journals/JIS/VOL21/Oteles/ote5.html">Jacobsthal Numbers and Associated Hessenberg Matrices</a>, J. Int. Seq., Vol. 21 (2018), Article 18.2.5.
%H A001045 D. Panario, M. Sahin, and Q. Wang, <a href="http://www.emis.de/journals/INTEGERS/papers/n78/n78.Abstract.html">A family of Fibonacci-like conditional sequences</a>, INTEGERS, Vol. 13, 2013, #A78.
%H A001045 D. Panario, M. Sahin, Q. Wang, and W. Webb, <a href="https://doi.org/10.1016/j.amc.2014.05.108">General conditional recurrences</a>, Applied Mathematics and Computation, Volume 243, Sep 15 2014, Pages 220-231.
%H A001045 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001045 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001045 M. Rahmani, <a href="https://doi.org/10.1016/j.laa.2012.07.050">The Akiyama-Tanigawa matrix and related combinatorial identities</a>, Linear Algebra and its Applications 438 (2013) 219-230. - From _N. J. A. Sloane_, Dec 26 2012
%H A001045 N. A. Rosenberg, <a href="https://doi.org/10.1534/genetics.115.181057">Admixture models and the breeding systems of H. S. Jennings: a GENETICS connection</a>, Genetics 202 (2016), 9-13.
%H A001045 E. Schlemm, <a href="http://arxiv.org/abs/1303.4979">On the expected number of successes in a sequence of nested Bernoulli trials</a>, arXiv preprint arXiv:1303.4979 [math.PR], 2013.
%H A001045 A. G. Shannon and J. V. Leyendekkers, <a href="http://nntdm.net/papers/nntdm-21/NNTDM-21-2-35-42.pdf">The Golden Ratio family and the Binet equation</a>, Notes on Number Theory and Discrete Mathematics, Vol. 21, 2015, No. 2, 35-42.
%H A001045 N. J. A. Sloane, <a href="https://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.
%H A001045 Yüksel Soykan, <a href="https://doi.org/10.9734/AIR/2019/v20i230154">On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers</a>, Advances in Research (2019) Vol. 20, No. 2, 1-15, Article AIR.51824.
%H A001045 Yüksel Soykan, <a href="https://doi.org/10.9734/jamcs/2020/v35i130241">Generalized Fibonacci Numbers: Sum Formulas</a>, Journal of Advances in Mathematics and Computer Science (2020) Vol. 35, No. 1, 89-104.
%H A001045 Yüksel Soykan, <a href="https://doi.org/10.9734/AJARR/2020/v9i130212">Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers</a>, Asian Journal of Advanced Research and Reports (2020) Vol. 9, No. 1, 23-39, Article no. AJARR.55441.
%H A001045 Yüksel Soykan, <a href="https://doi.org/10.34198/ejms.4220.297331">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</a>, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 297-331.
%H A001045 Yüksel Soykan, Erkan Taşdemir, and İnci Okumuş, <a href="https://doi.org/10.13140/RG.2.2.13499.36641">On Dual Hyperbolic Numbers With Generalized Jacobsthal Numbers Components</a>, Zonguldak Bülent Ecevit University, (Zonguldak, Turkey, 2019).
%H A001045 Paul K. Stockmeyer, <a href="https://arxiv.org/abs/1504.04404">The Pascal Rhombus and the Stealth Configuration</a>, arXiv:1504.04404 [math.CO], 2015.
%H A001045 X. Sun and V. H. Moll, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Moll/moll.html">The p-adic Valuations of Sequences Counting Alternating Sign Matrices</a>, JIS 12 (2009) 09.3.8.
%H A001045 Hassan Tarfaoui, <a href="http://d.tarfaoui.free.fr/cg/1991/EX4/exobis.pdf">Concours Général 1991 - Exercice 4</a> (in French).
%H A001045 Two-Year College Math. Jnl., <a href="http://www.jstor.org/stable/2687194">Review of "Jacobsthal Representation Numbers"</a>, 28 (1997), p. 76.
%H A001045 USA Mathematical Olympiad 2013, <a href="http://web.archive.org/web/20130606075948/http://amc.maa.org/usamo/2013/2013USAMO_Day1_Day2_Final_S.pdf">Problem 2</a> (proposed by Sam Vandervelde).
%H A001045 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobsthalNumber.html">Jacobsthal Number</a>, <a href="https://mathworld.wolfram.com/Negabinary.html">Negabinary</a>, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit</a>, <a href="https://mathworld.wolfram.com/Rule28.html">Rule 28</a>.
%H A001045 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>.
%H A001045 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>.
%H A001045 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>.
%H A001045 Wikipedia, <a href="http://en.wikipedia.org/wiki/Elementary_cellular_automaton">Elementary cellular automaton</a>
%H A001045 OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>.
%H A001045 Abdelmoumène Zekiri, Farid Bencherif, and Rachid Boumahdi, <a href="https://www.emis.de/journals/JIS/VOL21/Zekiri/zekiri4.html">Generalization of an Identity of Apostol</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.
%H A001045 G. B. M. Zerr, <a href="http://www.jstor.org/stable/2970981">Problem 64</a>, American Mathematical Monthly, vol. 3, no. 12, 1896 (p. 311).
%H A001045 <a href="/index/Cor#core">Index entries for "core" sequences</a>.
%H A001045 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).
%H A001045 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H A001045 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%H A001045 <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical competitions</a>.
%F A001045 a(n) = 2^(n-1) - a(n-1). a(n) = 2*a(n-1) - (-1)^n = (2^n - (-1)^n)/3.
%F A001045 G.f.: x/(1 - x - 2*x^2) = x/((x+1)*(1-2*x)). _Simon Plouffe_ in his 1992 dissertation
%F A001045 E.g.f.: (exp(2*x) - exp(-x))/3.
%F A001045 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
%F A001045 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
%F A001045 a(n) = Sum_{k=1..n} binomial(n, k)*(-1)^(n+k)*3^(k-1). - _Paul Barry_, Apr 02 2003
%F A001045 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
%F A001045 a(n) = U(n-1, i/(2*sqrt(2)))*(-i*sqrt(2))^(n-1) with i^2=-1. - _Paul Barry_, Nov 17 2003
%F A001045 a(n+1) = Sum_{k=0..ceiling(n/2)} 2^k*binomial(n-k, k). - _Benoit Cloitre_, Mar 06 2004
%F A001045 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
%F A001045 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
%F A001045 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
%F A001045 a(n+1) = Sum_{k=0..n} binomial(k, n-k)*2^(n-k). - _Paul Barry_, Oct 07 2004
%F A001045 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
%F A001045 From _Paul Barry_, Jan 17 2005: (Start)
%F A001045 a(n) = Sum_{k=0..n} k*binomial(n-1, (n-k)/2)*(1+(-1)^(n+k))*floor((2*k+1)/3).
%F A001045 a(n+1) = Sum_{k=0..n} k*binomial(n-1, (n-k)/2)*(1+(-1)^(n+k))*(A042965(k)+0^k). (End)
%F A001045 From _Paul Barry_, Jan 17 2005: (Start)
%F A001045 a(n+1) = ceiling(2^n/3) + floor(2^n/3) = (ceiling(2^n/3))^2 - (floor(2^n/3))^2.
%F A001045 a(n+1) = A005578(n) + A000975(n-1) = A005578(n)^2 - A000975(n-1)^2. (End)
%F A001045 a(n+1) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*binomial(j, k). - _Paul Barry_, Jan 26 2005
%F A001045 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
%F A001045 a(n) = ceiling(2^(n+1)/3) - ceiling(2^n/3) = A005578(n+1) - A005578(n). - _Paul Barry_, Oct 08 2005
%F A001045 a(n) = floor(2^(n+1)/3) - floor(2^n/3) = A000975(n) - A000975(n-1). - _Paul Barry_, Oct 08 2005
%F A001045 From _Paul Barry_, Feb 20 2003: (Start)
%F A001045 a(n) = Sum_{k=0..floor(n/3)} binomial(n, f(n-1)+3*k);
%F A001045 a(n) = Sum_{k=0..floor(n/3)} binomial(n, f(n-2)+3*k), where f(n)=A080425(n). (End)
%F A001045 From _Miklos Kristof_, Mar 07 2007: (Start)
%F A001045 a(2*n) = (1/3)*Product_{d|n} cyclotomic(d,4).
%F A001045 a(2*n+1) = (1/3)*Product_{d|2*n+1} cyclotomic(2*d,2). (End)
%F A001045 From _Hieronymus Fischer_, Apr 23 2007: (Start)
%F A001045 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}.
%F A001045 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
%F A001045 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
%F A001045 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}.
%F A001045 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)).
%F A001045 There is a similar formula regarding the arccos function, this is a(n) = 2^n/Pi*arccos(b(n)/2).
%F A001045 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))).
%F A001045 (End)
%F A001045 Sum_{k=0..n} A039599(n,k)*a(k) = A049027(n), for n >= 1. - _Philippe Deléham_, Jun 10 2007
%F A001045 Sum_{k=0..n} A039599(n,k)*a(k+1) = A067336(n). - _Philippe Deléham_, Jun 10 2007
%F A001045 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
%F A001045 a(n) + a(n+5) = 11*2^n. - _Paul Curtz_, Jan 17 2008
%F A001045 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
%F A001045 a(n) + a(n+2*k+1) = a(2*k+1)*2^n. - _Paul Curtz_, Feb 12 2008
%F A001045 a(n) = lower left term in the 2 X 2 matrix [0,2; 1,1]^n. - _Gary W. Adamson_, Mar 02 2008
%F A001045 a(n+1) = Sum_{k=0..n} A109466(n,k)*(-2)^(n-k). -_Philippe Deléham_, Oct 26 2008
%F A001045 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
%F A001045 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
%F A001045 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
%F A001045 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
%F A001045 For n > 1, a(n) = A000975(n-1) + (1 + (-1)^(n-1))/2. - _Vladimir Shevelev_, Feb 27 2012
%F A001045 From _Sergei N. Gladkovskii_, Jun 12 2012: (Start)
%F A001045 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).
%F A001045 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)
%F A001045 a(n) = 2^k * a(n-k) + (-1)^(n+k)*a(k). - _Paul Curtz_, _Jean-François Alcover_, Dec 11 2012
%F A001045 a(n) = sqrt((A014551(n))^2 + (-1)^(n-1)*2^(n+2))/3. - _Vladimir Shevelev_, Mar 13 2013
%F A001045 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
%F A001045 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
%F A001045 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
%F A001045 a(n+2) = Sum_{k=0..n} A108561(n,k)*(-2)^k. - _Philippe Deléham_, Nov 17 2013
%F A001045 a(n) = (Sum_{k=1..n, k odd} C(n,k)*3^(k-1))/2^(n-1). - _Vladimir Shevelev_, Feb 05 2014
%F A001045 a(-n) = -(-1)^n * a(n) / 2^n for all n in Z. - _Michael Somos_, Mar 18 2014
%F A001045 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
%F A001045 From _Peter Bala_, Apr 06 2015: (Start)
%F A001045 a(2*n)/a(n) = A014551(n) for n >= 1; a(3*n)/a(n) = 3*A245489(n) for n >= 1.
%F A001045 exp( Sum_{n >= 1} a(2*n)/a(n)*x^n/n ) = Sum_{n >= 0} a(n+1)*x^n.
%F A001045 exp( Sum_{n >= 1} a(3*n)/a(n)*x^n/n ) = Sum_{n >= 0} A084175(n+1)*x^n.
%F A001045 exp( Sum_{n >= 1} a(4*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015266(n+3)*(-x)^n.
%F A001045 exp( Sum_{n >= 1} a(5*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015287(n+4)*x^n.
%F A001045 exp( Sum_{n >= 1} a(6*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015305(n+5)*(-x)^n.
%F A001045 exp( Sum_{n >= 1} a(7*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015323(n+6)*x^n.
%F A001045 exp( Sum_{n >= 1} a(8*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015338(n+7)*(-x)^n.
%F A001045 exp( Sum_{n >= 1} a(9*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015356(n+8)*x^n.
%F A001045 exp( Sum_{n >= 1} a(10*n)/a(n)*x^n/n ) = Sum_{n >= 0} A015371(n+9)*(-x)^n. (End)
%F A001045 a(n) = (1-(-1)^n)/2 + floor((2^n)/3). - _Reiner Moewald_, Jun 05 2015
%F A001045 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
%F A001045 Dirichlet g.f.: (PolyLog(s,2) + (1 - 2^(1-s))*zeta(s))/3. - _Ilya Gutkovskiy_, Jun 27 2016
%F A001045 From _Yuchun Ji_, Apr 08 2018: (Start)
%F A001045 a(m)*a(n) + a(m-1)*a(n-1) - 2*a(m-2)*a(n-2) = 2^(m+n-3).
%F A001045 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).
%F A001045 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)
%F A001045 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
%F A001045 a(n) mod 10 = A091084(n). - _Alois P. Heinz_, Apr 25 2019
%F A001045 The sequence starting with "1" is the second INVERT transform of (1, -1, 3, -5, 11, -21, 43, ...). - _Gary W. Adamson_, Jul 08 2019
%F A001045 From _Kai Wang_, Jan 14 2020: (Start)
%F A001045 a(n)^2 - a(n+1)*a(n-1) = (-2)^(n-1).
%F A001045 a(n)^2 - a(n+r)*a(n-r) = (-2)^(n-r)*a(r)^2.
%F A001045 a(m)*a(n+1) - a(m+1)*a(n) = (-2)^n*a(m-n).
%F A001045 a(m-n) = (-1)^n*(a(m)*A014551(n) - A014551(m)*a(n))/(2^(n+1)).
%F A001045 a(m+n) = (a(m)*A014551(n) + A014551(m)*a(n))/2.
%F A001045 A014551(n)^2 - A014551(n+r)*A014551(n-r) = 9*(-1)^(n-r-1)*2^(n-r)*a(r)^2 .
%F A001045 A014551(m)*A014551(n+1) - A014551(m+1)*A014551(n) = 9*(-1)^(n-1)*2^(n)*a(m-n).
%F A001045 A014551(m-n) = (-1)^(n)*(A014551(m)*A014551(n) - 9*a(m)*a(n))/2^(n+1).
%F A001045 A014551(m+n) = (A014551(m)*A014551(n) + 9*a(m)*a(n))/2.
%F A001045 a(n) = Sum_{i=0..n-1; j=0..n-1; i+2*j=n-1} 2^j*((i+j)!/(i!*j!)). (End)
%F A001045 For n > 0, 1/(2*a(n+1)) = Sum_{m>=n} a(m)/(a(m+1)*a(m+2)). - _Kai Wang_, Mar 03 2020
%F A001045 For 4 > h >= 0, k >= 0, a(4*k+h) mod 5 = a(h) mod 5. - _Kai Wang_, May 07 2020
%F A001045 From _Kengbo Lu_, Jul 27 2020: (Start)
%F A001045 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.
%F A001045 a(n) = F(n) + Sum_{k=0..n-2} a(k)*F(n-k-1), where F denotes the Fibonacci numbers.
%F A001045 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).
%F A001045 a(n) = 1 + 2*Sum_{k=0..n-2} a(k).
%F A001045 a(m+n) = a(m)*a(n+1) + 2*a(m-1)*a(n).
%F A001045 a(2*n) = Sum_{i>=0, j>=0} binomial(n-j-1,i)*binomial(n-i-1,j)*2^(i+j). (End)
%F A001045 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
%F A001045 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
%F A001045 From _Peter Bala_, Jun 27 2025: (Start)
%F A001045 The following are all examples of telescoping infinite products:
%F A001045 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).
%F A001045 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).
%F A001045 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).
%F A001045 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)
%e A001045 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).
%e A001045 From _Joerg Arndt_, Nov 10 2012: (Start)
%e A001045 The a(6)=21 length-5 ternary words with no two consecutive letters nonzero are (dots for 0's)
%e A001045 [ 1] [ . . . . ]
%e A001045 [ 2] [ . . . 1 ]
%e A001045 [ 3] [ . . . 2 ]
%e A001045 [ 4] [ . . 1 . ]
%e A001045 [ 5] [ . . 2 . ]
%e A001045 [ 6] [ . 1 . . ]
%e A001045 [ 7] [ . 1 . 1 ]
%e A001045 [ 8] [ . 1 . 2 ]
%e A001045 [ 9] [ . 2 . . ]
%e A001045 [10] [ . 2 . 1 ]
%e A001045 [11] [ . 2 . 2 ]
%e A001045 [12] [ 1 . . . ]
%e A001045 [13] [ 1 . . 1 ]
%e A001045 [14] [ 1 . . 2 ]
%e A001045 [15] [ 1 . 1 . ]
%e A001045 [16] [ 1 . 2 . ]
%e A001045 [17] [ 2 . . . ]
%e A001045 [18] [ 2 . . 1 ]
%e A001045 [19] [ 2 . . 2 ]
%e A001045 [20] [ 2 . 1 . ]
%e A001045 [21] [ 2 . 2 . ]
%e A001045 (End)
%e A001045 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 + ...
%p A001045 A001045 := proc(n)
%p A001045 (2^n-(-1)^n)/3 ;
%p A001045 end proc: # _R. J. Mathar_, Dec 18 2012
%t A001045 Jacob0[n_] := (2^n - (-1)^n)/3; Table[Jacob0[n], {n, 0, 33}] (* _Robert G. Wilson v_, Dec 05 2005 *)
%t A001045 Array[(2^# - (-1)^#)/3 &, 33, 0] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
%t A001045 LinearRecurrence[{1, 2}, {0, 1}, 40] (* _Harvey P. Dale_, Nov 30 2011 *)
%t A001045 CoefficientList[Series[x/(1 - x - 2 x^2), {x, 0, 34}], x] (* _Robert G. Wilson v_, Jul 21 2015 *)
%t A001045 Table[(2^n - (-1)^n)/3, {n, 0, 20}] (* _Eric W. Weisstein_, Sep 21 2017 *)
%t A001045 Table[Abs[QBinomial[n, 1, -2]], {n, 0, 35}] (* _John Keith_, Jan 29 2022 *)
%o A001045 (PARI) a(n) = (2^n - (-1)^n) / 3
%o A001045 (PARI) 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
%o A001045 (PARI) a=0; for(n=0,34,print1(a,", "); a=2*(a-n%2)+1) \\ _K. Spage_, Aug 22 2014
%o A001045 (Sage) [lucas_number1(n, 1, -2) for n in range(34)] # _Zerinvary Lajos_, Apr 22 2009
%o A001045 # Alternatively:
%o A001045 a = BinaryRecurrenceSequence(1,2)
%o A001045 [a(n) for n in (0..34)] # _Peter Luschny_, Aug 29 2016
%o A001045 (Haskell)
%o A001045 a001045 = (`div` 3) . (+ 1) . a000079
%o A001045 a001045_list = 0 : 1 :
%o A001045 zipWith (+) (map (2 *) a001045_list) (tail a001045_list)
%o A001045 -- _Reinhard Zumkeller_, Mar 24 2013, Jan 05 2012, Feb 05 2011
%o A001045 (Maxima)
%o A001045 a[0]:0$
%o A001045 a[n]:=2^(n-1)-a[n-1]$
%o A001045 A001045(n):=a[n]$
%o A001045 makelist(A001045(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o A001045 (Python) # A001045.py
%o A001045 def A001045():
%o A001045 a, b = 0, 1
%o A001045 while True:
%o A001045 yield a
%o A001045 a, b = b, b+2*a
%o A001045 sequence = A001045()
%o A001045 [next(sequence) for i in range(20)] # _David Radcliffe_, Jun 26 2016
%o A001045 (Magma) [n le 2 select n-1 else Self(n-1)+2*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Jun 27 2016
%o A001045 (Python) [(2**n-(-1)**n)//3 for n in range(40)] # _Gennady Eremin_, Mar 03 2022
%Y A001045 Partial sums of this sequence give A000975, where there are additional comments from B. E. Williams and _Bill Blewett_ on the tie problem.
%Y A001045 A002487(a(n)) = A000045(n).
%Y A001045 Row sums of A059260, A156667 and A134317. Equals A026644(n-2)+1 for n > 1.
%Y A001045 a(n) = A073370(n-1, 0), n >= 1 (first column of triangle).
%Y A001045 Cf. A266508 (binary), A081857 (base 4), A147612 (characteristic function).
%Y A001045 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.
%Y A001045 Cf. A049883 = primes in this sequence, A107036 = indices of primes, A129738.
%Y A001045 Cf. A091084 (mod 10), A239473, A280049.
%Y A001045 Bisections: A002450, A007583.
%Y A001045 Cf. A077925 (signed version).
%K A001045 nonn,nice,easy,core,changed
%O A001045 0,4
%A A001045 _N. J. A. Sloane_
%E A001045 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