A085449 -id:A085449 - cp's OEIS frontend

A000351 Powers of 5: a(n) = 5^n.

1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 762939453125, 3814697265625, 19073486328125, 95367431640625, 476837158203125, 2384185791015625, 11920928955078125

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 5), L(1, 5), P(1, 5), T(1, 5). Essentially same as Pisot sequences E(5, 25), L(5, 25), P(5, 25), T(5, 25). See A008776 for definitions of Pisot sequences.
a(n) has leading digit 1 if and only if n = A067497 - 1. - Lekraj Beedassy, Jul 09 2002
With interpolated zeros 0, 1, 0, 5, 0, 25, ... (g.f.: x/(1 - 5*x^2)) second inverse binomial transform of Fibonacci(3n)/Fibonacci(3) (A001076). Binomial transform is A085449. - Paul Barry, Mar 14 2004
Sums of rows of the triangles in A013620 and A038220. - Reinhard Zumkeller, May 14 2006
Sum of coefficients of expansion of (1 + x + x^2 + x^3 + x^4)^n. a(n) is number of compositions of natural numbers into n parts less than 5. a(2) = 25 there are 25 compositions of natural numbers into 2 parts less than 5. - Adi Dani, Jun 22 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 5-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Numbers n such that sigma(5n) = 5n + sigma(n). In fact we have this theorem: p is a prime if and only if all solutions of the equation sigma(p*x) = p*x + sigma(x) are powers of p. - Jahangeer Kholdi, Nov 23 2013
From Doug Bell, Jun 22 2015: (Start)
Empirical observation: Where n is an odd multiple of 3, let x = (a(n) + 1)/9 and let y be the decimal expansion of x/a(n); then y*(x+1)/x + 1 = y rotated to the left.
Example:
  a(3) = 125;
  x = (125 + 1)/9 = 14;
  y = 112, which is the decimal expansion of 14/125 = 0.112;
  112*(14 + 1)/14 + 1 = 121 = 112 rotated to the left.
(End)
a(n) is the number of n-digit integers that contain only odd digits (A014261). - Bernard Schott, Nov 12 2022
Number of pyramids in the Sierpinski fractal square-based pyramid at the n-th step, while A279511 gives the corresponding number of vertices (see IREM link with drawings). - Bernard Schott, Nov 29 2022

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
O. M. Cain, The Exceptional Selfcondensability of Powers of Five, arXiv:1910.13829 [math.HO], 2019.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 270
IREM Paris-Nord, La pyramide de Sierpinski (in French).
Tanya Khovanova, Recursive Sequences
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
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Box Fractal
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A009969 (even bisection), A013710 (odd bisection), A005054 (first differences), A003463 (partial sums).
Cf. A006495, A006496, A159991, A001021.
Cf. A001076, A008776, A013620, A038220, A067497, A085449, A014261.
Sierpinski fractal square-based pyramid: A020858 (Hausdorff dimension), A279511 (number of vertices), this sequence (number of pyramids).

a000351 = (5 ^)
a000351_list = iterate (* 5) 1  -- Reinhard Zumkeller, Oct 31 2012

[5^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 27 2016

[ seq(5^n,n=0..30) ];
A000351:=-1/(-1+5*z); # Simon Plouffe in his 1992 dissertation

Table[5^n, {n, 0, 30}] (* Stefan Steinerberger, Apr 06 2006 *)
5^Range[0, 30] (* Harvey P. Dale, Aug 22 2011 *)

makelist(5^n,n,0,20); /* Martin Ettl, Dec 27 2012 */

a(n)=5^n \\ Charles R Greathouse IV, Jun 10 2011

def a(n): return 5**n
print([a(n) for n in range(24)]) # Michael S. Branicky, Nov 12 2022

(List.fill(50)(5: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, May 31 2019

a(n) = 5^n.
a(0) = 1; a(n) = 5*a(n-1) for n > 0.
G.f.: 1/(1 - 5*x).
E.g.f.: exp(5*x).
a(n) = A006495(n)^2 + A006496(n)^2.
a(n) = A159991(n) / A001021(n). - Reinhard Zumkeller, May 02 2009
From Bernard Schott, Nov 12 2022: (Start)
Sum_{n>=0} 1/a(n) = 5/4.
Sum_{n>=0} (-1)^n/a(n) = 5/6. (End)
a(n) = Sum_{k=0..n} C(2*n+1,n-k)*A000045(2*k+1). - Vladimir Kruchinin, Jan 14 2025

A015521 a(n) = 3a(n-1) + 4a(n-2), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861, 3355443, 13421773, 53687091, 214748365, 858993459, 3435973837, 13743895347, 54975581389, 219902325555, 879609302221, 3518437208883, 14073748835533

Offset: 0

Olivier Gérard

Inverse binomial transform of powers of 5 (A000351) preceded by 0. - Paul Barry, Apr 02 2003
Number of walks of length n between any two distinct vertices of the complete graph K_5. Example: a(2)=3 because the walks of length 2 between the vertices A and B of the complete graph ABCDE are: ACB, ADB, AEB. - Emeric Deutsch, Apr 01 2004
The terms of the sequence are the number of segments (sides) per iteration of the space-filling Peano-Hilbert curve. - Giorgio Balzarotti, Mar 16 2006
General form: k=4^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
A further inverse binomial transform generates A015441. - Paul Curtz, Nov 01 2009
For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 3's along the central diagonal, and 2's along the subdiagonal and the superdiagonal. - John M. Campbell, Jul 19 2011
Pisano period lengths: 1, 1, 2, 2, 10, 2, 6, 2, 6, 10, 10, 2, 6, 6, 10, 2, 4, 6, 18, 10, ... - R. J. Mathar, Aug 10 2012
Sum_{i=0..m} (-1)^(m+i)*4^i, for m >= 0, gives the terms after 0. - Bruno Berselli, Aug 28 2013
The ratio a(n+1)/a(n) converges to 4 as n approaches infinity. - Felix P. Muga II, Mar 09 2014
This is the Lucas sequence U(P=3,Q=-4), and hence for n>=0, a(n+2)/a(n+1) equals the continued fraction 3 + 4/(3 + 4/(3 + 4/(3 + ... + 4/3))) with n 4's. - Greg Dresden, Oct 07 2019
For n > 0, gcd(a(n), a(n+1)) = 1. - Kengbo Lu, Jul 27 2020

			G.f. = x + 3*x^2 + 13*x^3 + 51*x^4 + 205*x^5 + 819*x^6 + 3277*x^7 + 13107*x^8 + ...

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.
Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, and Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
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=3, b=2.
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
E. M. García-Caballero, S. G. Moreno, and M. P. Prophet, A complete view of Viète-like infinite products with Fibonacci and Lucas numbers, Applied Mathematics and Computation 247 (2014) 703-711.
Dale Gerdemann, Fractal generated from (3,4) recursion A015521, YouTube Video, Dec 4, 2014.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. A001045, A015518, A054878, A078008, A097073, A109200, A115341, A201455, A213127, A247281.

[Floor(4^n/5-(-1)^n/5): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(4^n/5),n=0..25) # Mircea Merca, Dec 28 2010

k=0;lst={k};Do[k=4^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
LinearRecurrence[{3,4}, {0,1}, 30] (* Harvey P. Dale, Jun 26 2012 *)
CoefficientList[Series[x/((1 - 4 x) (1 + x)), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)

a(n) = 4^n/5-(-1)^n/5; \\ Altug Alkan, Jan 08 2016

first(n) = Vec(x/(1 - 3*x - 4*x^2) + O(x^n), -n) \\ Iain Fox, Dec 30 2017

def A015521(n): return ((1<<(n<<1))|1)//5 # Chai Wah Wu, Jun 28 2023

[lucas_number1(n,3,-4) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009

From Paul Barry, Apr 02 2003: (Start)
a(n) = (4^n - (-1)^n)/5.
E.g.f.: (exp(4*x) - exp(-x))/5. (End)
a(n) = Sum_{k=1..n} binomial(n, k)*(-1)^(n+k)*5^(k-1). - Paul Barry, May 13 2003
a(2*n) = 4*a(2*n-1) - 1, a(2*n+1) = 4*a(2*n) + 1. In general this is true for all sequences of the type a(n) + a(n+1) = q^(n): i.e., a(2*n) = q*a(2n-1) - 1 and a(2*n+1) = q*a(2*n) + 1. - Amarnath Murthy, Jul 15 2003
From Emeric Deutsch, Apr 01 2004: (Start)
a(n) = 4^(n-1) - a(n-1).
G.f.: x/(1-3*x - 4*x^2). (End)
a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*3^(n-2k)*4^k. - Paul Barry, Jul 29 2004
a(n) = 4*a(n-1) - (-1)^n, n > 0, a(0)=0. - Paul Barry, Aug 25 2004
a(n) = Sum_{k=0..n} A155161(n,k)*2^(n-k), n >= 1. - Philippe Deléham, Jan 27 2009
a(n) = round(4^n/5). - Mircea Merca, Dec 28 2010
The logarithmic generating function 1/5*log((1+x)/(1-4*x)) = x + 3*x^2/2 + 13*x^3/3 + 51*x^4/4 + ... has compositional inverse 5/(4+exp(-5*x)) - 1, the e.g.f. for a signed version of A213127. - Peter Bala, Jun 24 2012
a(n) = (-1)^(n-1)*Sum_{k=0..n-1} A135278(n-1,k)*(-5)^k = (4^n - (-1)^n)/5 = (-1)^(n-1)*Sum_{k=0..n-1} (-4)^k. Equals (-1)^(n-1)*Phi(n,-4), where Phi is the cyclotomic polynomial when n is an odd prime. (For n > 0.) - Tom Copeland, Apr 14 2014
a(n+1) = 2^(2*n) - a(n), a(0) = 0. - Ben Paul Thurston, Dec 25 2015
a(n) = A247281(n)/5. - Altug Alkan, Jan 08 2016
From Kengbo Lu, Jul 27 2020: (Start)
a(n) = 3*Sum_{k=0..n-1} a(k) + 1 if n odd; a(n) = 3*Sum_{k=0..n-1} a(k) if n even.
a(n) = A030195(n) + Sum_{k=0..n-2} a(k)*A030195(n-k-1).
a(n) = A085449(n) + Sum_{k=0..n-1} a(k)*A085449(n-k).
a(n) = F(n) + 2*Sum_{k=0..n-1} a(k)*F(n-k) + 3*Sum_{k=0..n-2} a(k)*F(n-k-1), where F(n) denotes the Fibonacci numbers.
a(n) = F(n) + Sum_{k=0..n-1} a(k)*(L(n-k) + F(n-k+1)), where F(n) denotes the Fibonacci numbers and L(n) denotes the Lucas numbers.
a(n) = 3^(n-1) + 4*Sum_{k=0..n-2} 3^(n-k-2)*a(k).
a(m+n) = a(m)*a(n+1) + 4*a(m-1)*a(n).
a(2*n) = Sum_{i>=0, j>=0} binomial(n-j-1,i)*binomial(n-i-1,j)*3^(2n-2i-2j-1)*4^(i+j). (End)

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

A103435 a(n) = 2^n * Fibonacci(n).

Original entry on oeis.org

0, 2, 4, 16, 48, 160, 512, 1664, 5376, 17408, 56320, 182272, 589824, 1908736, 6176768, 19988480, 64684032, 209321984, 677380096, 2192048128, 7093616640, 22955425792, 74285318144, 240392339456, 777925951488, 2517421260800

Offset: 0

Ralf Stephan, Feb 08 2005

Cardinality of set of bracelets of size at most n that are tiled with two types of colored squares and four types of colored dominoes.
a(n) is also the diagonal element of the matrix A(i,j) whose first row (i=1) and first column (j=1) are the Fibonacci numbers: A(1,k)=A(k,1)=fib(k) and whose generic element is the sum of element in adjacent (preceding) row and column minus the absolute value of their difference. So a(n) = A(n,n) = A(i-1,j)+A(i,j-1)-abs(A(i-1,j)-A(i,j-1)). - Carmine Suriano, May 13 2010
a(n) is the coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) given for d=sqrt(x+1) by p(n,x)=((x+d)^n-(x-d)^n)/(2d), for n>=1.  The constant terms under this reduction are the absolute values of terms of A086344.  See A192232 for a discussion of reduction. - Clark Kimberling, Jun 29 2011
The exponential convolution of A000032 and A000045. - Vladimir Reshetnikov, Oct 06 2016

			a(5)=160=A(5,5)=A(4,5)+A(5,4)-abs[A(4,5)+A(5,4)]=80+80-0. - _Carmine Suriano_, May 13 2010
G.f. = 2*x + 4*x^2 + 16*x^3 + 48*x^4 + 160*x^5 + 512*x^6 + 1664*x^7 + ...

Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, identity 236, p. 131.

Tom Edgar, Extending Some Fibonacci-Lucas Relations, The Fibonacci Quarterly, Vol. 54, No. 1 (2016), p. 79.
Harris Kwong, An Alternate Proof of Sury's Fibonacci-Lucas Relation, The American Mathematical Monthly, Vol. 121, No. 6 (2014), p. 514.
Diego Marques, A new Fibonacci-Lucas relation, Amer. Math. Monthly, Vol. 122, No. 7 (2015), p. 683.
Ivica Martinjak and Helmut Prodinger, Complementary Families of the Fibonacci-Lucas Relations, arXiv:1508.04949 [math.CO], 2015-2016.
B. Sury, A polynomial parent to a Fibonacci-Lucas relations, Amer. Math. Monthly, Vol. 121, No. 3 (2014), p. 236.
Charles R. Wall, Problem B-607, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), p. 370; Product of Exponential Generating Functions, Solution to Problem B-607 by Bob Prielipp, ibid., Vol. 26, No. 4 (1988), pp. 374-375.
Index entries for linear recurrences with constant coefficients, signature (2,4).

First differences of A014334.
Partial sums of A087131.
Cf. A000032, A000045, A006483, A063727, A085449, A269991.

[2^n *Fibonacci(n): n in [0..50]]; // Vincenzo Librandi, Apr 04 2011

Expand[Table[((1 + Sqrt[5])^n - (1 - Sqrt[5])^n)5/(5 Sqrt[5]), {n, 0, 25}]] (* Zerinvary Lajos, Mar 22 2007 *)
Table[2^n Fibonacci[n],{n,0,40}] (* or *) LinearRecurrence[{2,4},{0,2},40] (* Harvey P. Dale, Oct 14 2020 *)

a(n)=fibonacci(n)<Charles R Greathouse IV, Feb 03 2014

concat(0, Vec(2*x/(1-2*x-4*x^2) + O(x^99))) \\ Altug Alkan, May 11 2016

a(n) = A006483(n) + 1 = 2*A085449(n) = 2*A063727(n-1), n>0.
G.f.: 2*x / (1 - 2*x - 4*x^2).
a(n) = Sum_{i=0..n-1}( 2^i * Lucas(i) ).
a(n) = 2*a(n-1) + 4*a(n-2). - Carmine Suriano, May 13 2010
a(n) = a(-n) * -(-4)^n for all n in Z. - Michael Somos, Sep 20 2014
E.g.f.: 2*sinh(sqrt(5)*x)*exp(x)/sqrt(5). - Ilya Gutkovskiy, May 10 2016
Sum_{n>=1} 1/a(n) = (1/2) * A269991. - Amiram Eldar, Nov 17 2020
a(n) == 2*n (mod 10). - Amiram Eldar, Jan 15 2022
a(n) = Sum_{k=0..n} binomial(n,k) * Fibonacci(k) * Lucas(n-k) (Wall, 1987). - Amiram Eldar, Jan 27 2022

A234357 Array T(n,k) by antidiagonals: T(n,k) = n^k * Fibonacci(k).

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 18, 24, 5, 5, 32, 81, 80, 8, 6, 50, 192, 405, 256, 13, 7, 72, 375, 1280, 1944, 832, 21, 8, 98, 648, 3125, 8192, 9477, 2688, 34, 9, 128, 1029, 6480, 25000, 53248, 45927, 8704, 55, 10, 162, 1536, 12005, 62208, 203125, 344064, 223074, 28160, 89, 11, 200, 2187

Offset: 0

Ralf Stephan, Dec 24 2013

			Array starts:
1,  2,   3,    5,     8,     13,    21,   34, 55, 89,...    (A000045)
2,  8,  24,   80,   256,    832,  2688, 8704,...   (A063727, A085449)
3, 18,  81,  405,  1944,   9477, 45927,...         (A122069, A099012)
4, 32, 192, 1280,  8192,  53248,...                         (A099133)
5, 50, 375, 3125, 25000, 203125,...
6, 72, 648, 6480, 62208, 606528,...
...
Columns: A000027, A001105, A117642.

PARI
```
T(n,k)=n^k*fibonacci(k)
```

T(n,k)=polcoeff(Ser(1/(1-n*x-n^2*x^2)),k)

G.f. of n-th row: 1/(1 - n*x - n^2*x^2).

Recurrence: T(n,k) = n*T(n,k-1) + n^2*T(n,k-2), starting n, 2*n^2.

A269991 Decimal expansion of Sum_{n >= 1} 2^(1-n)/Fibonacci(n).

Original entry on oeis.org

1, 6, 8, 4, 8, 1, 3, 1, 4, 4, 4, 8, 9, 5, 7, 6, 0, 9, 6, 3, 1, 6, 5, 5, 4, 3, 3, 7, 3, 8, 0, 0, 7, 8, 2, 3, 0, 2, 3, 7, 0, 6, 3, 8, 8, 2, 4, 5, 7, 0, 8, 6, 8, 2, 0, 9, 4, 3, 1, 7, 6, 1, 8, 8, 5, 9, 5, 0, 5, 6, 0, 0, 2, 8, 0, 4, 9, 4, 5, 4, 9, 8, 9, 1, 0, 8

Offset: 1

Clark Kimberling, Mar 12 2016

			1.684813144489576096316554337380078230...

Cf. A000045, A269992.

Cf. A063727, A085449, A103435, A209084.

x = N[Sum[2^(1 - n)/Fibonacci[n], {n, 1, 500}], 100]
RealDigits[x][[1]]

suminf(n=1, 2^(1-n)/fibonacci(n)) \\ Michel Marcus, Feb 01 2021

Equals Sum_{n>=0} 1/A063727(n) = Sum_{n>=1} 1/A085449(n) = 2 * Sum_{n>=1} 1/A103435(n) = 4 * Sum_{n>=1} 1/A209084(n). - Amiram Eldar, Feb 01 2021

A087205 a(n) = -2a(n-1) + 4a(n-2), a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 0, 8, -16, 64, -192, 640, -2048, 6656, -21504, 69632, -225280, 729088, -2359296, 7634944, -24707072, 79953920, -258736128, 837287936, -2709520384, 8768192512, -28374466560, 91821703168, -297141272576, 961569357824, -3111703805952

Offset: 0

Paul Barry, Aug 25 2003

Inverse binomial transform of A087204.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,4)

Cf. A000045, A063727, A084222, A085449, A087204.

[(-1)^(n+1)*2^n*Fibonacci(n-2): n in [0..50]]; // G. C. Greubel, Oct 08 2018

Table[-(-2)^n*Fibonacci[n - 2], {n, 0, 50}] (* G. C. Greubel, Oct 08 2018 *)
LinearRecurrence[{-2,4},{1,2},30] (* Harvey P. Dale, Jan 24 2022 *)

Vec((4*x+1)/(-4*x^2+2*x+1)+O(x^66)) \\ Joerg Arndt, Jul 14 2013

vector(50, n, n--; (-1)^(n+1)*2^n*fibonacci(n-2)) \\ G. C. Greubel, Oct 08 2018

a(n) = (-1-sqrt(5))^n * (1/2-3*sqrt(5)/10) + (-1+sqrt(5))^n * (1/2+3*sqrt(5)/10).
G.f.: (4*x +1)/(-4*x^2 +2*x +1). - Joerg Arndt, Jul 14 2013
a(n+2) = A085449(n)*(-1)^(n+1); a(n+3) = A063727(n)*(-1)^n.
a(n) = -(-2)^n*F(n-2) for n >= 0, with F = A000045, and F(-1) = 1, F(-2) = -1. - Wolfdieter Lang, Oct 08 2018

A123004 Expansion of g.f. x^2/(1 - 2x - 25x^2).

Original entry on oeis.org

0, 1, 2, 29, 108, 941, 4582, 32689, 179928, 1177081, 6852362, 43131749, 257572548, 1593438821, 9626191342, 59088353209, 358831489968, 2194871810161, 13360530869522, 81592856993069, 497198985724188, 3034219396275101

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, 25).

Sequences of the form (m*i)^(n-2)*ChebyshevU(n-2, -i/m): A131577 (m=0), A000129 (m=1), A085449 (m=2), A002534 (m=3), A161007 (m=4), this sequence (m=5), A123005 (m=7), A123006 (m=11).

[n le 2 select n-1 else 2*Self(n-1) +25*Self(n-2): n in [1..30]]; // G. C. Greubel, Jul 12 2021

Rest@CoefficientList[Series[x^2/(1 -2*x -25*x^2), {x,0,40}], x]
Join[{a=0,b=1},Table[c=2*b+25*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)

[(5*i)^(n-2)*chebyshev_U(n-2, -i/5) for n in [1..30]] # G. C. Greubel, Jul 12 2021

a(n) = 2*a(n-1) + 25*a(n-2).
a(n+1) = ((1+sqrt(26))^n - (1-sqrt(26))^n)/(2*sqrt(26)). - Rolf Pleisch, Jul 06 2009
a(n) = (5*i)^(n-2)*ChebyshevU(n-2, -i/5). - G. C. Greubel, Jul 12 2021

Definition replaced by generating function - the Assoc. Eds. of the OEIS, Mar 27 2010

A123005 Expansion of g.f. x^2/(1-2x-49x^2).

Original entry on oeis.org

0, 1, 2, 53, 204, 3005, 16006, 179257, 1142808, 11069209, 78136010, 698663261, 5225991012, 44686481813, 345446523214, 2880530655265, 22687940948016, 186521884004017, 1484752874460818, 12109078065118469, 96971046978817020

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,49).

Sequences of the form (m*i)^(n-1)*ChebyshevU(n-1, -i/m): A131577 (m=0), A000129 (m=1), A085449 (m=2), A002534 (m=3), A161007 (m=4), A123004 (m=5), this sequence (m=7), A123006 (m=11).

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) -49*Self(n-2): n in [1..31]]; // G. C. Greubel, Jul 12 2021

CoefficientList[Series[x^2/(1-2x-49x^2),{x,0,30}],x] (* Harvey P. Dale, Apr 12 2020 *)

[(7*i)^(n-2)*chebyshev_U(n-2, -i/7) for n in [1..30]] # G. C. Greubel, Jul 12 2021

a(n) = 2*a(n-1) + 49*a(n-2).

a(n) = (7*i)^(n-2)*ChebyshevU(n-2, -i/7). - G. C. Greubel, Jul 12 2021

Definition replaced by generating function - the Assoc. Eds. of the OEIS, Mar 27 2010

A123006 Expansion of x^2/(1 -2x -121x^2).

Original entry on oeis.org

0, 1, 2, 125, 492, 16109, 91750, 2132689, 15367128, 288789625, 2437001738, 39817548101, 374512306500, 5566947933221, 56449884952942, 786500469825625, 8403437018957232, 111973430886815089, 1240762741067455250

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..800
Index entries for linear recurrences with constant coefficients, signature (2, 121).

Sequences of the form (m*i)^(n-1)*ChebyshevU(n-1, -i/m): A131577 (m=0), A000129 (m=1), A085449 (m=2), A002534 (m=3), A161007 (m=4), A123004 (m=5), A123005 (m=7), this sequence (m=11).

[n le 2 select n-1 else 2*Self(n-1) + 121*Self(n-2): n in [1..30]]; // G. C. Greubel, Jul 12 2021

Rest@CoefficientList[Series[x^2/(1 -2*x -121*x^2), {x,0,30}], x]

[(11*i)^(n-2)*chebyshev_U(n-2, -i/11) for n in [1..30]] # G. C. Greubel, Jul 12 2021

a(n) = 2*a(n-1) + 121*a(n-2).

a(n) = (11*i)^(n-2)*ChebyshevU(n-2, -i/11). - G. C. Greubel, Jul 12 2021

cp's OEIS Frontend

A000351 Powers of 5: a(n) = 5^n.

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

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A234357 Array T(n,k) by antidiagonals: T(n,k) = n^k * Fibonacci(k).

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A269991 Decimal expansion of Sum_{n >= 1} 2^(1-n)/Fibonacci(n).

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A015521 a(n) = 3a(n-1) + 4a(n-2), a(0) = 0, a(1) = 1.

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

A087205 a(n) = -2a(n-1) + 4a(n-2), a(0)=1, a(1)=2.

A123004 Expansion of g.f. x^2/(1 - 2x - 25x^2).

A123005 Expansion of g.f. x^2/(1-2x-49x^2).

A123006 Expansion of x^2/(1 -2x -121x^2).