A006131 -id:A006131 - cp's OEIS frontend

A006130 a(n) = a(n-1) + 3*a(n-2) for n > 1, a(0) = a(1) = 1.

1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, 6160, 14209, 32689, 75316, 173383, 399331, 919480, 2117473, 4875913, 11228332, 25856071, 59541067, 137109280, 315732481, 727060321, 1674257764, 3855438727, 8878212019, 20444528200, 47079164257, 108412748857

Offset: 0

N. J. A. Sloane

Counts walks of length n at the vertex of degree five in the graph with adjacency matrix A = [1,1,1,1;1,0,0,0;1,0,0,0;1,0,0,0]. - Paul Barry, Oct 02 2004
Form the graph with matrix A = [0,1,1,1;1,1,0,0;1,0,1,0;1,0,0,1]. The sequence 0,1,1,4,... counts walks of length n between the vertex without loop and another vertex. - Paul Barry, Oct 02 2004
Length-n words with letters {0,1,2,3} where no two consecutive letters are nonzero, see fxtbook link below. - Joerg Arndt, Apr 08 2011
Hankel transform is the sequence [1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 10 2007
Let M = [1, sqrt(3); sqrt(3), 0] be a 2 X 2 matrix. Then A006130(n)={[M^n]A006130-A052533%20=%20A006130%20(shifted%20to%20the%20right%20one%20place,%20with%20first%20term%20=%200).%20-%20_L.%20Edson%20Jeffery">(1,1)}. Note that A006130-A052533 = A006130 (shifted to the right one place, with first term = 0). - _L. Edson Jeffery, Nov 25 2011 [Any matrix M = [1, y; 3/y, 0], with y not 0, will do it. - Wolfdieter Lang, Feb 18 2018]
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, 4*a(n-2) equals the number of 4-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011
a(n) is the number of compositions (ordered partitions) of n into parts 1 and 2 where there are three sorts of part 2 (see the g.f.). - Joerg Arndt, Jan 16 2024
Number of pairs of rabbits when there are 3 pairs per litter and offspring reach parenthood after 2 gestation periods. - Robert FERREOL, Oct 28 2018
Numerators of stationary probabilities sequence for number of customers in steady state of M2/M/1 queue, whose g.f. is (1-z)/(3-3z-z(1-z^2)). - Igor Kleiner, Nov 03 2018
INVERT transform of (1, 0, 3, 0, 9, 0, 27, ...). - Gary W. Adamson, Jul 15 2019
Number of 3-compositions of n+2 with 1 not allowed as a part; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 17 2020
Number of ways to tile a strip of length n with 3 colors of dominoes and 1 color of squares. - Greg Dresden, Sep 01 2021
Number of 3-permutations of n elements avoiding the patterns 231, 312, 321. See Bonichon and Sun. - Michel Marcus, Aug 20 2022
a(n-1), with a(-1) = 0, appears in the formula for the powers of the fundamental (integer) algebraic number c = (1 + sqrt(13))/2 = A209927: c^n = A052533(n) + a(n-1)*c. With the formulas given below, and also in A052533, in terms of S-Chebyshev polynomials this is valid also for the powers of 1/c = (-1 + sqrt(13))/6 = A356033. - Wolfdieter Lang, Nov 26 2023

			G.f. = 1 + x + 4*x^2 + 7*x^3 + 19*x^4 + 40*x^5 + 97*x^6 + 217*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Stephen Wolfram, 'The Mathematica Book,' Fourth Edition, Wolfram Media or Cambridge University Press, 1999, p. 96.

Vincenzo Librandi, G. C. Greubel and Robert Israel, Table of n, a(n) for n = 0..2485 (n = 0..149 from Vincenzo Librandi, n = 150..300 from G. C. Greubel)
Joerg Arndt, Matters Computational (The Fxtbook), pp.317-318.
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 436
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Thor Martinsen, Non-Fisherian generalized Fibonacci numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 2, 370-389. See pp. 387-389.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Arulalan Rajan, R. Vittal Rao, Ashok Rao and H. S. Jamadagni, Fibonacci Sequence, Recurrence Relations, Discrete Probability Distributions and Linear Convolution, arXiv preprint arXiv:1205.5398 [math.PR], 2012.
A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, Vol. 21, 2015, No. 2, 35-42.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
A. K. Whitford, Binet's formula generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.
Fatih Yılmaz and Mustafa Özkan, On the Generalized Gaussian Fibonacci Numbers and Horadam Hybrid Numbers: A Unified Approach, Axioms (2022) Vol. 11, No. 6, 255.
Paul Thomas Young, p-adic congruences for generalized Fibonacci sequences, The Fibonacci Quarterly, Vol. 32, No. 1, 1994.
Index entries for linear recurrences with constant coefficients, signature (1,3).
Index entries for sequences related to Chebyshev polynomials.

Cf. A006131, A015440, A049310, A052533, A140167, A175291 (Pisano periods), A099232 (partial sums), A274977.

Cf. A209927, A356033.

a := [1, 1];; for n in [3..30] do a[n] := a[n-1] + 3*a[n-2]; od; a; # Muniru A Asiru, Feb 18 2018

[n le 2 select 1 else Self(n-1) + 3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 17 2012

a := n -> add(binomial(n-k, k)*3^k, k=0..n): seq(a(n), n=0..29); # Zerinvary Lajos, Sep 30 2006
f:= gfun:-rectoproc({a(n) = a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 1},a(n),remember):
map(f, [$0..100]); # Robert Israel, Aug 31 2015

a[0] = a[1] = 1; a[n_] := a[n] = a[n - 1] + 3a[n - 2]; Table[ a[n], {n, 0, 30}]
LinearRecurrence[{1, 3}, {1, 1}, 30] (* Vincenzo Librandi, Oct 17 2012 *)
RecurrenceTable[{a[n]== a[n-1] + 3*a[n-2], a[0]== 1, a[1]== 1}, a, {n,0,30}] (* G. C. Greubel, Aug 30 2015 *)
a[0] := 1; a[n_] := Hypergeometric2F1[1/2-n/2, -n/2, -n, -12]; Table[a[n], {n, 0, 29}] (* Peter Luschny, Feb 18 2018 *)
a[ n_] := With[{s = Sqrt[-1/3]}, ChebyshevU[n, s/2] / s^n] // Simplify; (* Michael Somos, Nov 04 2018 *)

Vec(1/(1-x-3*x^2+O(x^66))) \\ Franklin T. Adams-Watters, May 26 2011

an = an1 = 1
while an<10**5:
   print(an)
   an1 += an*3
   an = an1 - an*3   # Alex Ratushnyak, Apr 20 2012

from sage.combinat.sloane_functions import recur_gen2
it = recur_gen2(1,1,1,3)
[next(it) for i in range(30)] # Zerinvary Lajos, Jun 25 2008

[lucas_number1(n,1,-3) for n in range(1, 31)] # Zerinvary Lajos, Apr 22 2009

O.g.f.: 1/(1 - x - 3*x^2). - Simon Plouffe in his 1992 dissertation.
a(n) = (( (1 + sqrt(13))/2 )^(n+1) - ( (1 - sqrt(13))/2 )^(n+1))/sqrt(13).
a(n) = Sum_{k = 0..ceiling(n/2)} 3^k*C(n-k, k). - Benoit Cloitre, Philippe Deléham, Mar 07 2004
a(0) = 1; a(1) = 1; for n >= 1, a(n+1) = (a(n)^2 - (-3)^n) / a(n-1). - Philippe Deléham, Mar 07 2004
The i-th term of the sequence is the (1, 2) entry in the i-th power of the 2 X 2 matrix M = ((-1, 1), (1, 2)). - Simone Severini, Oct 15 2005
a(n) = lower right term in the 2 X 2 matrix [0,3; 1,1]^n. - Gary W. Adamson, Mar 02 2008
a(n) = Sum_{k = 0..n} A109466(n,k)*(-3)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = Product_{k = 1..floor((n - 1)/2)} (1 + 12*cos(k*Pi/n)^2). - Roger L. Bagula and Gary W. Adamson, Nov 21 2008
Limiting ratio = (1 + sqrt(13))/2 = 2.30277563.. = A098316 - 1. - Roger L. Bagula and Gary W. Adamson, Nov 21 2008
G.f.: G(0)/(2-x), where G(k)= 1 + 1/(1 - x*(13*k - 1)/(x*(13*k + 12) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 18 2013
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + 3*x)/( x*(4*k+3 + 3*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013
a(n) = ( Sum_{1 <= k <= n+1, k odd} C(n+1,k)*13^((k-1)/2) )/2^n. - Vladimir Shevelev, Feb 05 2014
E.g.f.: (1/(a - b))*(a*exp(a*x) - b*exp(b*x)), where 2*a = 1 + sqrt(13) and 2*b = 1 - sqrt(13). - G. C. Greubel, Aug 30 2015
a(n) = ((i*sqrt(3))^n)*S(n, (-i/sqrt(3))), with the imaginary unit i and the Chebyshev S polynomials (coefficients in A049310). - Wolfdieter Lang, Feb 18 2018
a(n) = hypergeom([(1-n)/2, -n/2], [-n], -12) for n >= 1. - Peter Luschny, Feb 18 2018
a(n) = 3 * (-3)^n * a(-2-n) for all n in Z. - Michael Somos, Nov 04 2018
a(n) = ( sqrt(3) )^n * Fibonacci(n+1, 1/sqrt(3)). - G. C. Greubel, Dec 26 2019
a(n) = numerator of the continued fraction 1 + 3/(1 + 3/(1 + 3/ ... + 3/1)) with exactly n 1's, for n>0. - Greg Dresden and Alexander Mark, Aug 16 2020
With an initial 0 prepended, the sequence [0, 1, 1, 4, 7, 19, 40, ...] satisfies the congruences a(n*p^k) == e*a(n*p^(k-1)) (mod p^k) for positive integers k and n and all primes p, where e = +1 for the primes p listed in A296937, e = 0 if p = 13, otherwise e = -1. See Young, Theorem 1, Corollary 1 (i). - Peter Bala, Dec 28 2022
From Wolfdieter Lang, Nov 27 2023: (Start)
a(n) = sqrt(-3)^n*S(n, 1/sqrt(-3)) with the S-Chebyshev polynomials (see A049310), also valid for negative n, using S(-n, x) = -S(n-2, x), for n >= 2, and S(-1, x) = 0. See above the formula in terms of Fibonacci polynomials).
a(n) = A052533(n+2)/3, for n >= 0. (End)
From Peter Bala, Jun 27 2025: (Start)
G.f.: Sum_{n >= 0} x^n * Product_{k = 1..n} (k + 3*x)/(1 + k*x) = Sum_{n >= 0} x^n * Product_{k = 1..n} (1 + 3*k*x)/(1 + 3*k*x^2).
The following products telescope:
Product_{k >= 0} (1 + 3^k/a(2*k+1)) = 1 + sqrt(13).
Product_{k >= 1} (1 - 3^k/a(2*k+1)) = 1/14 * (1 + sqrt(13)).
Product_{k >= 0} (1 + (-3)^k/a(2*k+1)) = (1/13) * (13 + sqrt(13)).
Product_{k >= 1} (1 - (-3)^k/a(2*k+1)) = (1/14) * (13 + sqrt(13)). (End)

A109466 Riordan array (1, x(1-x)).

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 0, -2, 1, 0, 0, 1, -3, 1, 0, 0, 0, 3, -4, 1, 0, 0, 0, -1, 6, -5, 1, 0, 0, 0, 0, -4, 10, -6, 1, 0, 0, 0, 0, 1, -10, 15, -7, 1, 0, 0, 0, 0, 0, 5, -20, 21, -8, 1, 0, 0, 0, 0, 0, -1, 15, -35, 28, -9, 1, 0, 0, 0, 0, 0, 0, -6, 35, -56, 36, -10, 1, 0, 0, 0, 0, 0, 0, 1, -21, 70, -84, 45, -11, 1, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Aug 28 2005

Inverse is Riordan array (1, xc(x)) (A106566).
Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, -1, 1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008
Coefficient array of the polynomials Chebyshev_U(n, sqrt(x)/2)*(sqrt(x))^n. - Paul Barry, Sep 28 2009

			Rows begin:
  1;
  0,  1;
  0, -1,  1;
  0,  0, -2,  1;
  0,  0,  1, -3,  1;
  0,  0,  0,  3, -4,   1;
  0,  0,  0, -1,  6,  -5,   1;
  0,  0,  0,  0, -4,  10,  -6,   1;
  0,  0,  0,  0,  1, -10,  15,  -7,  1;
  0,  0,  0,  0,  0,   5, -20,  21, -8,  1;
  0,  0,  0,  0,  0,  -1,  15, -35, 28, -9, 1;
From _Paul Barry_, Sep 28 2009: (Start)
Production array is
  0,    1,
  0,   -1,    1,
  0,   -1,   -1,   1,
  0,   -2,   -1,  -1,   1,
  0,   -5,   -2,  -1,  -1,  1,
  0,  -14,   -5,  -2,  -1, -1,  1,
  0,  -42,  -14,  -5,  -2, -1, -1,  1,
  0, -132,  -42, -14,  -5, -2, -1, -1,  1,
  0, -429, -132, -42, -14, -5, -2, -1, -1, 1 (End)

Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
Tom Copeland, Addendum to Elliptic Lie Triad

Cf. A026729 (unsigned version), A000108, A030528, A124644.

/* As triangle */ [[(-1)^(n-k)*Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jan 14 2016

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, #(1-#)&, 13] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Number triangle T(n, k) = (-1)^(n-k)*binomial(k, n-k).
T(n, k)*2^(n-k) = A110509(n, k); T(n, k)*3^(n-k) = A110517(n, k).
Sum_{k=0..n} T(n,k)*A000108(k)=1. - Philippe Deléham, Jun 11 2007
From Philippe Deléham, Oct 30 2008: (Start)
Sum_{k=0..n} T(n,k)*A144706(k) = A082505(n+1).
Sum_{k=0..n} T(n,k)*A002450(k) = A100335(n).
Sum_{k=0..n} T(n,k)*A001906(k) = A100334(n).
Sum_{k=0..n} T(n,k)*A015565(k) = A099322(n).
Sum_{k=0..n} T(n,k)*A003462(k) = A106233(n). (End)
Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = -12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12 respectively. - Philippe Deléham, Oct 27 2008
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n+1), A057086(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively. - Philippe Deléham, Oct 28 2008
G.f.: 1/(1-y*x+y*x^2). - Philippe Deléham, Dec 15 2011
T(n,k) = T(n-1,k-1) - T(n-2,k-1), T(n,0) = 0^n. - Philippe Deléham, Feb 15 2012
Sum_{k=0..n} T(n,k)*x^(n-k) = F(n+1,-x) where F(n,x)is the n-th Fibonacci polynomial in x defined in A011973. - Philippe Deléham, Feb 22 2013
Sum_{k=0..n} T(n,k)^2 = A051286(n). - Philippe Deléham, Feb 26 2013
Sum_{k=0..n} T(n,k)*T(n+1,k) = -A110320(n). - Philippe Deléham, Feb 26 2013
For T(0,0) = 0, the signed triangle below has the o.g.f. G(x,t) = [t*x(1-x)]/[1-t*x(1-x)] = L[t*Cinv(x)] where L(x) = x/(1-x) and Cinv(x)=x(1-x) with the inverses Linv(x) = x/(1+x) and C(x)= [1-sqrt(1-4*x)]/2, an o.g.f. for the shifted Catalan numbers A000108, so the inverse o.g.f. is Ginv(x,t) = C[Linv(x)/t] = [1-sqrt[1-4*x/(t(1+x))]]/2 (cf. A124644 and A030528). - Tom Copeland, Jan 19 2016

A063727 a(n) = 2a(n-1) + 4a(n-2), a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 8, 24, 80, 256, 832, 2688, 8704, 28160, 91136, 294912, 954368, 3088384, 9994240, 32342016, 104660992, 338690048, 1096024064, 3546808320, 11477712896, 37142659072, 120196169728, 388962975744, 1258710630400

Offset: 0

Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Aug 12 2001

Essentially the same as A085449.
Convergents to 2*golden ratio = (1+sqrt(5)).
Number of ways to tile an n-board with two types of colored squares and four types of colored dominoes.
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 5 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(5). - Cino Hilliard, Sep 25 2005
a(n) is also the quasi-diagonal element A(i-1,i)=A(1,i-1) of matrix A(i,j) whose elements in first row A(1,k) and first column A(k,1) equal k-th Fibonacci Fib(k) and the generic element is the sum of adjacent (previous) in row and column minus the absolute value of their difference. - Carmine Suriano, May 13 2010
Equals INVERT transform of A006131: (1, 1, 5, 9, 29, 65, 181, ...). - Gary W. Adamson, Aug 12 2010
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 2's along the three central diagonals. - John M. Campbell, Jul 19 2011
The numbers composing the denominators of the fractional limit to A134972. - Seiichi Kirikami, Mar 06 2012
Pisano period lengths:  1, 1, 8, 1, 5, 8, 48, 1, 24, 5, 10, 8, 42, 48, 40, 1, 72, 24, 18, 5, ... - R. J. Mathar, Aug 10 2012

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 235.
John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Harry J. Smith)
J. Borowska and L. Lacinska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=2, b=2.
Index entries for linear recurrences with constant coefficients, signature (2,4).
Index entries for sequences related to Chebyshev polynomials.

Cf. A006483, A103435, A006131, A269991.
Second row of A234357. Row sums of triangle A016095.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

List([0..25],n->2^n*Fibonacci(n+1)); # Muniru A Asiru, Nov 24 2018

[n le 2 select n else 2*Self(n-1) + 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 07 2018

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+4*a[n-2]od: seq(a[n], n=1..33); # Zerinvary Lajos, Dec 15 2008

a[n_]:=(MatrixPower[{{1,5},{1,1}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
CoefficientList[Series[1/(1 - 2 x - 4 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 31 2014 *)
LinearRecurrence[{2, 4}, {1, 2}, 50] (* G. C. Greubel, Jan 07 2018 *)

s(n)=if(n<2,n+1,(s(n-1)+(s(n-2)*2))*2); for(n=0,32,print(s(n)))

{ for (n=0, 200, if (n>1, a=2*a1 + 4*a2; a2=a1; a1=a, if (n, a=a1=2, a=a2=1)); write("b063727.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 28 2009

[lucas_number1(n,2,-4) for n in range(1, 26)] # Zerinvary Lajos, Apr 22 2009

a(n) = 2 * A087206(n+1).
From Vladeta Jovovic, Aug 16 2001: (Start)
a(n) = sqrt(5)/10*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1)).
G.f.: 1/(1-2*x-4*x^2). (End)
From Mario Catalani (mario.catalani(AT)unito.it), Jun 13 2003: (Start)
a(2*n) = 4*a(n-1)^2 + a(n)^2.
A084057(n+1)/a(n) converges to sqrt(5). (End)
E.g.f.: exp(x)*(cosh(sqrt(5)*x)+sinh(sqrt(5)*x)/sqrt(5)). - Paul Barry, Sep 20 2003
a(n) = 2^n*Fibonacci(n+1). - Vladeta Jovovic, Oct 25 2003
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)*5^k. - Paul Barry, Nov 15 2003
a(n) = U(n, i/2)*(-i*2)^n, i^2=-1. - Paul Barry, Nov 17 2003
Simplified formula: ((1+sqrt(5))^n-(1-sqrt(5))^n)/sqrt(20). Offset 1. a(3)=8. - Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009
First binomial transform of 1,1,5,5,25,25. - Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009
a(n) = A(n-1,n) = A(n,n-1); A(i,j) = A(i-1,j) + A(i,j-1) - abs(A(i-1,j) - A(i,j-1)). - Carmine Suriano, May 13 2010
G.f.: G(0) where G(k) = 1 + 2*x*(1+2*x)/(1 - 2*x*(1+2*x)/(2*x*(1+2*x) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 31 2013
G.f.: G(0)/(2*(1-x)), where G(k) = 1 + 1/(1 - x*(5*k-1)/(x*(5*k+4) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+2 + 4*x )/( x*(4*k+4 + 4*x ) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Sep 21 2013
Sum_{n>=0} 1/a(n) = A269991. - Amiram Eldar, Feb 01 2021

Better description from Jason Earls and Vladeta Jovovic, Aug 16 2001

Incorrect comment removed by Greg Dresden, Jun 02 2020

A026597 Expansion of (1+x)/(1-x-4*x^2).

Original entry on oeis.org

1, 2, 6, 14, 38, 94, 246, 622, 1606, 4094, 10518, 26894, 68966, 176542, 452406, 1158574, 2968198, 7602494, 19475286, 49885262, 127786406, 327327454, 838473078, 2147782894, 5501675206, 14092806782, 36099507606, 92470734734

Offset: 0

Clark Kimberling

This sequence can generated by the following formula: a(n) = a(n-1) + 4*a(n-2) when n > 2; a[1] = 1, a[2] = 2. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 21 2004
An elephant sequence, see A175654 and A175655. For the corner squares just one A[5] vector, with decimal value 325, leads to the sequence given above. For the central square this vector leads to a companion sequence that is 4 times this very same sequence with n >= -1. - Johannes W. Meijer, Aug 15 2010
Equals INVERTi transform of A180168. - Gary W. Adamson, Aug 14 2010
Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 2 X 2 cells and remove the cells that have one '1' in their modulo 3 coordinates. a(n) is the number of cells after n iterations. Cell configuration converges to a fractal with approximate dimension 1.357. - Peter Karpov, Apr 20 2017
Also, the number of walks of length n starting at vertex 1 in the graph with 4 vertices and edges {{0,1}, {0,2}, {0,3}, {1,2}, {2,3}}. - Sean A. Irvine, Jun 02 2025

Nathaniel Johnston, Table of n, a(n) for n = 0..500
Sean A. Irvine, Walks on Graphs.
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..8)
Index entries for linear recurrences with constant coefficients, signature (1,4).

Cf. A006131, A006138, A007482, A026581, A026584, A122950, A175654, A175655, A180168.

[n le 2 select n else Self(n-1) + 4*Self(n-2): n in [1..41]]; // G. C. Greubel, Dec 08 2021

LinearRecurrence[{1,4},{1,2},40] (* Harvey P. Dale, Nov 28 2011 *)

[(2*i)^n*( chebyshev_U(n, -i/4) - (i/2)*chebyshev_U(n-1, -i/4) ) for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1+x)/(1-x-4*x^2).
a(n) = T(n,0) + T(n,1) + ... + T(n,2*n), T given by A026584.
a(n) = Sum_{k=0..n} binomial(floor((2*n-k-1)/2), n-k)*2^k. - Paul Barry, Feb 11 2005
a(n) = A006131(n) + A006131(n-1), n >= 1. - R. J. Mathar, Oct 20 2006
a(n) = Sum_{k=0..n} binomial(floor((2*n-k)/2),n-k)*4^floor(k/2). - Paul Barry, Feb 02 2007
Inverse binomial transform of A007482: (1, 3, 11, 39, 139, 495, ...). - Gary W. Adamson, Dec 04 2007
a(n) = Sum_{k=0..n+1} A122950(n+1,k)*3^(n+1-k). - Philippe Deléham, Jan 04 2008
a(n) = (1/2 + 3*sqrt(17)/34)*(1/2 + sqrt(17)/2)^n + (1/2 - 3*sqrt(17)/34)*(1/2 - sqrt(17)/2)^n. - Antonio Alberto Olivares, Jun 07 2011
a(n) = (2*i)^n*( chebyshevU(n, -i/4) - (i/2)*chebyshevU(n-1, -i/4) ). - G. C. Greubel, Dec 08 2021
E.g.f.: exp(x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, Jan 31 2023

Better name from Ralf Stephan, Jul 14 2013

A168561 Riordan array (1/(1-x^2), x/(1-x^2)). Unsigned version of A049310.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 1, 0, 3, 0, 4, 0, 1, 1, 0, 6, 0, 5, 0, 1, 0, 4, 0, 10, 0, 6, 0, 1, 1, 0, 10, 0, 15, 0, 7, 0, 1, 0, 5, 0, 20, 0, 21, 0, 8, 0, 1, 1, 0, 15, 0, 35, 0, 28, 0, 9, 0, 1, 0, 6, 0, 35, 0, 56, 0, 36, 0, 10, 0, 1, 1, 0, 21, 0, 70, 0, 84, 0, 45, 0, 11, 0, 1

Offset: 0

Philippe Deléham, Nov 29 2009

Row sums: A000045(n+1), Fibonacci numbers.
A168561*A007318 = A037027, as lower triangular matrices. Diagonal sums : A077957. - Philippe Deléham, Dec 02 2009
T(n,k) is the number of compositions of n+1 into k+1 odd parts. Example: T(4,2)=3 because we have 5 = 1+1+3 = 1+3+1 = 3+1+1.
Coefficients of monic Fibonacci polynomials (rising powers of x). Ftilde(n, x) = x*Ftilde(n-1, x) + Ftilde(n-2, x), n >=0, Ftilde(-1,x) = 0, Ftilde(0, x) = 1. G.f.: 1/(1 - x*z - z^2). Compare with Chebyshev S-polynomials (A049310). - Wolfdieter Lang, Jul 29 2014

			The triangle T(n,k) begins:
n\k 0  1   2   3   4    5    6    7    8    9  10  11  12  13 14 15 ...
0:  1
1:  0  1
2:  1  0   1
3:  0  2   0   1
4:  1  0   3   0   1
5:  0  3   0   4   0    1
6:  1  0   6   0   5    0    1
7:  0  4   0  10   0    6    0    1
8:  1  0  10   0  15    0    7    0    1
9:  0  5   0  20   0   21    0    8    0    1
10: 1  0  15   0  35    0   28    0    9    0   1
11: 0  6   0  35   0   56    0   36    0   10   0   1
12: 1  0  21   0  70    0   84    0   45    0  11   0   1
13: 0  7   0  56   0  126    0  120    0   55   0  12   0   1
14: 1  0  28   0 126    0  210    0  165    0  66   0  13   0  1
15: 0  8   0  84   0  252    0  330    0  220   0  78   0  14  0  1
... reformatted by _Wolfdieter Lang_, Jul 29 2014.
------------------------------------------------------------------------

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From _N. J. A. Sloane_, May 10 2012
Tom Copeland, Addendum to Elliptic Lie Triad
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A162515 (rows reversed), A112552, A102426 (deflated).

A168561:=proc(n,k) if n-k mod 2 = 0 then binomial((n+k)/2,k) else 0 fi end proc:
seq(seq(A168561(n,k),k=0..n),n=0..12) ; # yields sequence in triangular form

Table[If[EvenQ[n + k], Binomial[(n + k)/2, k], 0], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 16 2017 *)

T(n,k) = if ((n+k) % 2, 0, binomial((n+k)/2,k));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print();); \\ Michel Marcus, Oct 09 2016

Sum_{k=0..n} T(n,k)*x^k = A059841(n), A000045(n+1), A000129(n+1), A006190(n+1), A001076(n+1), A052918(n), A005668(n+1), A054413(n), A041025(n), A099371(n+1), A041041(n), A049666(n+1), A041061(n), A140455(n+1), A041085(n), A154597(n+1), A041113(n) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 respectively. - Philippe Deléham, Dec 02 2009
T(2n,2k) = A085478(n,k). T(2n+1,2k+1) = A078812(n,k). Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000045(n+1), A006131(n), A015445(n), A168579(n), A122999(n) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Dec 02 2009
T(n,k) = binomial((n+k)/2,k) if (n+k) is even; otherwise T(n,k)=0.
G.f.: (1-z^2)/(1-t*z-z^2) if offset is 1.
T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = 1, T(0,1) = 0. - Philippe Deléham, Feb 09 2012
Sum_{k=0..n} T(n,k)^2 = A051286(n). - Philippe Deléham, Feb 09 2012
From R. J. Mathar, Feb 04 2022: (Start)
Sum_{k=0..n} T(n,k)*k = A001629(n+1).
Sum_{k=0..n} T(n,k)*k^2 = 0,1,4,11,... = 2*A055243(n)-A099920(n+1).
Sum_{k=0..n} T(n,k)*k^3 = 0,1,8,29,88,236,... = 12*A055243(n) -6*A001629(n+2) +A001629(n+1)-6*(A001872(n)-2*A001872(n-1)). (End)

Typo in name corrected (1(1-x^2) changed to 1/(1-x^2)) by Wolfdieter Lang, Nov 20 2010

A015440 a(n) = a(n-1) + 5*a(n-2), with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 6, 11, 41, 96, 301, 781, 2286, 6191, 17621, 48576, 136681, 379561, 1062966, 2960771, 8275601, 23079456, 64457461, 179854741, 502142046, 1401415751, 3912125981, 10919204736, 30479834641, 85075858321, 237475031526, 662854323131, 1850229480761, 5164501096416

Offset: 0

Olivier Gérard

Original name: Generalized Fibonacci numbers.
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, 6*a(n-2) equals the number of 6-colored compositions of n with all parts >= 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011
Pisano period lengths: 1, 3, 6, 6, 1, 6, 21, 12, 18, 3, 40, 6, 56, 21, 6, 24, 16, 18, 360, 6, .... - R. J. Mathar, Aug 10 2012
From Wolfdieter Lang, Jan 02 2024: (Start)
This sequence {a(n-1)}, with a(-1) = 0, appears in the formula for powers of phi21 := (1 + sqrt(21))/2 = A222134 = 2.791287..., together with A(n) = A365824(n), as phi21^n = A(n) + a(n-1)*phi21(n), for n >= 0.
Limit_{n->oo} a(n)/a(n-1) = phi21. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Joerg Arndt, Matters Computational (The Fxtbook), section 14.8, pp. 317-318
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 2, (2015), 35-42.
Paul Thomas Young, p-adic congruences for generalized Fibonacci sequences, The Fibonacci Quarterly, Vol. 32, No. 1, 1994.
Index entries for linear recurrences with constant coefficients, signature (1,5).

Cf. A006130, A006131, A015441, A049310, A222134, A365824.

[n le 2 select 1 else Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 06 2012

A015440 := proc(n)
    if n <= 1 then
        1;
    else
        procname(n-1)+5*procname(n-2) ;
    end if;
end proc: # R. J. Mathar, May 15 2016

a[n_]:=(MatrixPower[{{1,3},{1,-2}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{1, 5}, {1, 1}, 100] (* Vincenzo Librandi, Nov 06 2012 *)

a(n)=abs([1,3;1,-2]^n*[1;1])[2,1] \\ Charles R Greathouse IV, Feb 03 2014

[lucas_number1(n,1,-5) for n in range(1, 28)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1) + 5*a(n-2).
a(n) = (( (1+sqrt(21))/2 )^(n+1) - ( (1-sqrt(21))/2 )^(n+1))/sqrt(21).
a(n) = Sum_{k=0..ceiling(n/2)} 5^k*binomial(n-k, k). - Benoit Cloitre, Mar 06 2004
G.f.: 1/(1 - x - 5x^2). - R. J. Mathar, Sep 03 2008
a(n) = Sum_{k=0..n} A109466(n,k)*(-5)^(n-k). - Philippe Deléham, Oct 26 2008
From Jeffrey R. Goodwin, May 28 2011: (Start)
A special case of a more general class of Lucas sequences given by
U(n) = U(n-1) + (4^(m-1)-1)/3 U(n-2).
U(n) = (( (1+sqrt((4^(m)-1)/3))/2 )^(n+1) - ( (1-sqrt((4^(m)-1)/3))/2 )^(n+1))/sqrt((4^(m)-1)/3). Fix m = 2 to get the formula for the Fibonacci sequence, fix m = 3 to get the formula for a(n). (End)
G.f.: G(0)/(2-x), where G(k)= 1 + 1/(1 - x*(21*k-1)/(x*(21*k+20) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
G.f.: Q(0)/x -1/x, where Q(k) = 1 + 5*x^2 + (k+2)*x - x*(k+1 + 5*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 06 2013
a(n) = (Sum_{k=1..n+1, k odd} binomial(n+1,k)*21^((k-1)/2))/2^n. - Vladimir Shevelev, Feb 05 2014
With an initial 0 prepended, the sequence [0, 1, 1, 6, 11, 41, 96, ...] satisfies the congruences a(n*p^k) == (3|p)*(7|p)*a(n*p^(k-1)) (mod p^k) for positive integers k and n and all primes p, where (n|p) denotes the Legendre symbol. See Young, Theorem 1, Corollary 1(i). - Peter Bala, Dec 28 2022
a(n) = sqrt(-5)^(n-1)*S(n-1,1/sqrt(-5)), for n >= 0, with the Chebyshev polynomial S(n, x) (see A049310). - Wolfdieter Lang, Nov 17 2023
From Peter Bala, Jun 27 2025: (Start)
The following products telescope:
Product_{k >= 0} (1 + 5^k/a(2*k+1)) = 1 + sqrt(21).
Product_{k >= 1} (1 - 5^k/a(2*k+1)) = 1/22 * (1 + sqrt(21)).
Product_{k >= 0} (1 + (-5)^k/a(2*k+1)) = (1/21) * (21 + sqrt(21)).
Product_{k >= 1} (1 - (-5)^k/a(2*k+1)) = (1/22) * (21 + sqrt(21)). (End)
E.g.f.: exp(x/2)*(sqrt(21)*cosh(sqrt(21)*x/2) + sinh(sqrt(21)*x/2))/sqrt(21). - Stefano Spezia, Jul 04 2025

A053404 Expansion of 1/((1+3x)(1-4*x)).

Original entry on oeis.org

1, 1, 13, 25, 181, 481, 2653, 8425, 40261, 141361, 624493, 2320825, 9814741, 37664641, 155441533, 607417225, 2472715621, 9761722321, 39434309773, 156574977625, 629786694901, 2508686426401, 10066126765213, 40170363882025

Offset: 0

Barry E. Williams, Jan 07 2000

Hankel transform is := 1,12,0,0,0,... - Philippe Deléham, Nov 02 2008

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, 13*a(n-2) equals the number of 13-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
A. K. Whitford, Binet's formula generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.
Index entries for linear recurrences with constant coefficients, signature (1,12).

Cf. A000045, A001045, A006130, A006131, A015440 - A015443, A015445 - A015447, A053404, A053428, A168579, A350468, A350469.

[((4^(n+1)) - (-3)^(n+1))/7: n in [0..30]]; // G. C. Greubel, Jan 16 2018

seq(simplify(hypergeom([1/2 - (1/2)*n, -(1/2)*n], [-n], -48)), n = 1..40); # Peter Bala, Jul 05 2025

CoefficientList[Series[1/((1 + 3 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 06 2014 *)

a(n)=([0,1; 12,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number1(n,1,-12) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

a(n) = ((4^(n+1))-(-3)^(n+1))/7.
a(n) = a(n-1) + 12*a(n-2), n > 1; a(0)=1, a(1)=1.
From Paul Barry, Jul 30 2004: (Start)
Convolution of 4^n and (-3)^n.
G.f.: 1/((1+3x)(1-4x)); a(n) = Sum_{k=0..n, 4^k*(-3)^(n-k)} = Sum_{k=0..n, (-3)^k*4^(n-k)}. (End)
a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*(-12)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = (sum_{1<=k<=n+1, k odd} C(n+1,k)*7^(k-1))/2^n. - Vladimir Shevelev, Feb 05 2014
From Peter Bala, Jun 27 2025: (Start)
a(n) = hypergeom([1/2 - (1/2)*n, -(1/2)*n], [-n], -48) for n >= 1.
The following products telescope:
Product_{k >= 0} (1 + 12^k/a(2*k+1)) = 8.
Product_{k >= 1} (1 - 12^k/a(2*k+1)) = 4/25.
Product_{k >= 0} (1 + (-12)^k/a(2*k+1)) = 8/7.
Product_{k >= 1} (1 - (-12)^k/a(2*k+1)) = 28/25. (End)

More terms from James Sellers, Feb 02 2000

A105309 a(n) = |b(n)|^2 = x^2 + 3y^2 where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)(0,c,c,c) where c = 1/sqrt(3).

Original entry on oeis.org

1, 1, 2, 5, 9, 20, 41, 85, 178, 369, 769, 1600, 3329, 6929, 14418, 30005, 62441, 129940, 270409, 562725, 1171042, 2436961, 5071361, 10553600, 21962241, 45703841, 95110562, 197926885, 411889609, 857150100, 1783745641, 3712008565

Offset: 0

Gerald McGarvey, Apr 25 2005

Prepending 0 and keeping the offset at 0, turns this into a divisibility sequence with g.f. x(1-x^2)/(1-x-2x^2-x^3+x^4). - T. D. Noe, Dec 22 2008
Equals INVERT transform of (1, 1, 2, 0, 2, 0, 2, ...). - Gary W. Adamson, Apr 28 2009
Sequence gives the norm of the coefficients in 1/(1 - I*x - I*x^2), where I^2=-1. - Paul D. Hanna, Dec 06 2011
This is the case P1 = 1, P2 = -4, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 27 2014

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 9*x^4 + 20*x^5 + 41*x^6 + 85*x^7 + 178*x^8 + ...

Peter Bala, Linear divisibility sequences and Chebyshev polynomials
Ricky X. F. Chen and Louis W. Shapiro, On Sequences G(n) satisfying G(n) = (d+2)G(n-1)-G(n-2), J. Int. Seq. 10 (2007) 07.8.1, Theorem 16.
Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 8.
Eric Weisstein's World of Mathematics, Quaternion
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (1,2,1,-1).

Cf. A092886, A201837, A201838.

Cf. A006131, A100047, A240513

a[ n_] := (ChebyshevT[n + 1, (1 + Sqrt[17])/4] - ChebyshevT[n + 1, (1 - Sqrt[17])/4]) 2 / Sqrt[17] // Simplify; (* Michael Somos, Dec 20 2016 *)

{a(n) = my(A); n = abs(n+1)-1; if( n<2, n>=0, n++; A = vector(n, i, 1); for(i=3, n, A[i] = A[i-1] + A[i-2]*I); norm(A[n]))}; /* Michael Somos, Apr 28 2005 */

{a(n)=norm(polcoeff(1/(1-I*x-I*x^2+x*O(x^n)), n))} /* Paul D. Hanna */

{a(n)=polcoeff((1-x^2)/(1-x-2*x^2-x^3+x^4)+x*O(x^n),n)}

a(n) = A092886(n+1) - A092886(n-1), n > 0.
a(n) = A201837(n)^2 + A201838(n)^2. - Paul D. Hanna, Dec 06 2011
From Peter Bala, Mar 27 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(17))/4 and beta = (1 - sqrt(17))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 1/2].
a(n) = U(n-1,(1 + i)/sqrt(8))*U(n-1,(1 - i)/sqrt(8)), where U(n,x) denotes the Chebyshev polynomial of the second kind.
The o.g.f. is the Chebyshev transform of the rational function x/(1 - x + 4*x^2) = x + x^2 + 5*x^2 + 9*x^4 + 29*x^5 + ... (see A006131), where the Chebyshev transform takes the function A(x) to the function (1 - x^2)/(1 + x^2)*A(x/(1 + x^2)).
See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
a(n) = abs(((sqrt(4*i - 1) + i)^(n+1) - (i - sqrt(4*i - 1))^(n+1)) / 2^(n+1) / sqrt(4*i - 1))^2. - Daniel Suteu, Dec 20 2016
a(n) = a(-2-n) for all n in Z. - Michael Somos, Dec 20 2016
G.f.: (1+x)*(1-x)/(1-x-2*x^2-x^3+x^4). - R. J. Mathar, Apr 26 2024

cp's OEIS Frontend

A072270 A partial product representation of A006131 and A072265.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A006130 a(n) = a(n-1) + 3*a(n-2) for n > 1, a(0) = a(1) = 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Python

Sage

Sage

Formula

A109466 Riordan array (1, x(1-x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A063727 a(n) = 2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

Extensions

A026597 Expansion of (1+x)/(1-x-4*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A168561 Riordan array (1/(1-x^2), x/(1-x^2)). Unsigned version of A049310.

Views

Author

Keywords

Comments

Examples

A063727 a(n) = 2a(n-1) + 4a(n-2), a(0)=1, a(1)=2.

A053404 Expansion of 1/((1+3x)(1-4*x)).

A105309 a(n) = |b(n)|^2 = x^2 + 3y^2 where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)(0,c,c,c) where c = 1/sqrt(3).