A084057 -id:A084057 - cp's OEIS frontend

A026150 a(0) = a(1) = 1; a(n+2) = 2a(n+1) + 2a(n).

1, 1, 4, 10, 28, 76, 208, 568, 1552, 4240, 11584, 31648, 86464, 236224, 645376, 1763200, 4817152, 13160704, 35955712, 98232832, 268377088, 733219840, 2003193856, 5472827392, 14952042496, 40849739776

Offset: 0

N. J. A. Sloane

a(n+1)/A002605(n) converges to sqrt(3). - Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003
a(n+1)/a(n) converges to 1 + sqrt(3) = 2.732050807568877293.... - Philippe Deléham, Jul 03 2005
Binomial transform of expansion of cosh(sqrt(3)x) (A000244 with interpolated zeros); inverse binomial transform of A001075. - Philippe Deléham, Jul 04 2005
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 3 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(3). - Cino Hilliard, Sep 25 2005
Inverse binomial transform of A001075: (1, 2, 7, 26, 97, 362, ...). - Gary W. Adamson, Nov 23 2007
Starting (1, 4, 10, 28, 76, ...), the sequence is the binomial transform of [1, 3, 3, 9, 9, 27, 27, 81, 81, ...], and inverse binomial transform of A001834: (1, 5, 19, 71, 265, ...). - Gary W. Adamson, Nov 30 2007
[1, 3; 1, 1]^n * [1,0] = [a(n), A002605(n)]. - Gary W. Adamson, Mar 21 2008
(1 + sqrt(3))^n = a(n) + A002605(n)*(sqrt(3)). - Gary W. Adamson, Mar 21 2008
Equals right border of triangle A143908. Also, starting (1, 4, 10, 28, ...) = row sums of triangle A143908 and INVERT transform of (1, 3, 3, 3, ...). - Gary W. Adamson, Sep 06 2008
a(n) is the number of compositions of n when there are 1 type of 1 and 3 types of other natural numbers. - Milan Janjic, Aug 13 2010
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 85, 277, 337 and 340, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A002605 (without the leading 0). - Johannes W. Meijer, Aug 15 2010
Pisano period lengths: 1, 1, 1, 1, 24, 1, 48, 1, 3, 24, 10, 1, 12, 48, 24, 1,144, 3,180, 24, ... - R. J. Mathar, Aug 10 2012
(1 + sqrt(3))^n = a(n) + A002605(n)*sqrt(3), for n >= 0; integers in the real quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 10 2018
a(n) is also the number of solutions for cyclic three-dimensional stable matching instances with master preference lists of size n (Escamocher and O'Sullivan 2018). - Guillaume Escamocher, Jun 15 2018
Starting from a(1), first differences of A005665. - Ivan N. Ianakiev, Nov 22 2019
Number of 3-permutations of n elements avoiding the patterns 231, 312. See Bonichon and Sun. - Michel Marcus, Aug 19 2022

			G.f. = 1 + x + 4*x^2 + 10*x^3 + 28*x^4 + 76*x^5 + 208*x^6 + 568*x^7 + ...

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8.
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
A. Burstein, S. Kitaev and T. Mansour, Independent sets in certain classes of (almost) regular graphs, arXiv:math/0310379 [math.CO], 2003.
Guillaume Escamocher and Barry O'Sullivan, Three-Dimensional Matching Instances Are Rich in Stable Matchings, CPAIOR 2018, pages 182-197.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1052
Tanya Khovanova, Recursive Sequences
Emanuele Munarini, A generalization of André-Jeannin's symmetric identity, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 98-118.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (2,2).
Index entries for sequences related to Chebyshev polynomials.

First differences of A002605.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Cf. A001075, A001834, A083337, A002605, A143908, A028859, A030195, A106435, A108898, A125145, A053120.

a026150 n = a026150_list !! n
a026150_list = 1 : 1 : map (* 2) (zipWith (+) a026150_list (tail
a026150_list))
-- Reinhard Zumkeller, Oct 15 2011

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

with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):ZL1:=Prod(begin_blockP, Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL2,ZL2,ZL2), b=ZL1], ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, {Q}, unlabelled], size=n)/3, n=2..27); # Zerinvary Lajos, Mar 08 2008

Expand[Table[((1 + Sqrt[3])^n + (1 - Sqrt[3])^n)/(2), {n, 0, 30}]] (* Artur Jasinski, Dec 10 2006 *)
LinearRecurrence[{2, 2}, {1, 1}, 30] (* T. D. Noe, Mar 25 2011 *)
Round@Table[LucasL[n, Sqrt[2]] 2^(n/2 - 1), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2016 *)

a(n) := if n<=1 then 1 else 2*a(n-1)+2*a(n-2);
makelist(a(n),n,0,20); /* Emanuele Munarini, Apr 14 2017 */

{a(n) = if( n<0, 0, real((1 + quadgen(12))^n))};

from sage.combinat.sloane_functions import recur_gen2; it = recur_gen2(1,1,2,2); [next(it) for i in range(30)] # Zerinvary Lajos, Jun 25 2008

[lucas_number2(n,2,-2)/2 for n in range(0, 26)] # Zerinvary Lajos, Apr 30 2009

a(n) = (1/2)*((1 + sqrt(3))^n + (1 - sqrt(3))^n). - Benoit Cloitre, Oct 28 2002
G.f.: (1 - x)/(1 - 2*x - 2*x^2).
a(n) = a(n-1) + A083337(n-1). A083337(n)/a(n) converges to sqrt(3). - Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003
From Paul Barry, May 15 2003: (Start)
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*3^k;
E.g.f.: exp(x)*cosh(sqrt(3)x). (End)
a(n) = Sum_{k=0..n} A098158(n,k)*3^(n - k). - Philippe Deléham, Dec 26 2007
a(n) = upper left and lower right terms of [1, 1; 3, 1]^n. (1 + sqrt(3))^n = a(n) + A083337(n)/(sqrt(3)). - Gary W. Adamson, Mar 12 2008
a(n) = A080040(n)/2. - Philippe Deléham, Nov 19 2008
If p[1] = 1, and p[i] = 3, (i > 1), and if A is 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) = det A. - Milan Janjic, Apr 29 2010
a(n) = 2 * A052945(n-1). - Vladimir Joseph Stephan Orlovsky, Mar 24 2011
a(n) = round((1 + sqrt(3))^n/2) for n > 0. - Bruno Berselli, Feb 04 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k - 1)/(x*(3*k + 2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
a(n) = (-sqrt(2)*i)^n*T(n,sqrt(2)*i/2), with i = sqrt(-1) and the Chebyshev T-polynomials (A053120). - Wolfdieter Lang, Feb 10 2018

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

A002532 a(n) = 2a(n-1) + 5a(n-2), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, 9, 28, 101, 342, 1189, 4088, 14121, 48682, 167969, 579348, 1998541, 6893822, 23780349, 82029808, 282961361, 976071762, 3366950329, 11614259468, 40063270581, 138197838502, 476712029909, 1644413252328, 5672386654201, 19566839570042, 67495612411089

Offset: 0

N. J. A. Sloane

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 6 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(6). - Cino Hilliard, Sep 25 2005
For n>=2, number of ordered partitions of n-1 into parts of sizes 1 and 2 where there are two types of 1 (singletons) and five types of 2 (twins). For example, the number of possible configurations of families of n-1 male (M) and female (F) offspring considering only single births and twins, where the birth order of M/F/pair-of-twins is considered and there are five types of twins; namely, both F (identical twins), both F (fraternal twins), both M (identical), both M (fraternal), or one F and one M - where birth order within a pair of twins itself is disregarded. In particular, for a(3)=9, two children could be either: (1) F, then M; (2) M, then F; (3) F,F; (4) M,M; (5) F,F identical twins; (6) F,F fraternal twins; (7) M,M identical twins; (8) M,M fraternal twins; or (9) M,F twins (emphasizing that birth order is irrelevant here when children are the same gender, when two children are within the same pair of twins and when pairs of twins have both the same gender(s) and identical-vs-fraternal characteristics). - Rick L. Shepherd, Sep 19 2004
Pisano period lengths: 1, 2, 3, 4, 4, 6, 24, 8, 3, 4, 120, 12, 56, 24, 12, 16, 288, 6, 18, 4, ... . - R. J. Mathar, Aug 10 2012

			G.f. = x + 2*x^2 + 9*x^3 + 28*x^4 + 101*x^5 + 342*x^6 + 1189*x^7 + ...

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
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).
A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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
Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12 [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,5).

Cf. A015581 (similar application, but no distinguishing identical vs. fraternal twins).

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

[Floor(((1+Sqrt(6))^n-(1-Sqrt(6))^n)/(2*Sqrt(6))): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

[n le 2 select n-1 else 2*Self(n-1) + 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

A002532:=-z/(-1+2*z+5*z**2); # Conjectured by Simon Plouffe in his 1992 dissertation
# second program
seq(simplify(2^(n-1) * hypergeom([1 - (1/2)*n, 1/2 - (1/2)*n], [1 - n], -5)), n = 2..25); # Peter Bala, Jul 06 2025

Expand[Table[((1 + Sqrt[6])^n - (1 - Sqrt[6])^n)/(2Sqrt[6]), {n, 0, 25}]] (* Zerinvary Lajos, Mar 22 2007 *)
a[n_]:=(MatrixPower[{{1,2},{1,-3}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{2,5},{0,1},30] (* Harvey P. Dale, Nov 03 2011 *)

Vec(1/(1-2*x-5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Apr 17 2012

from sage.combinat.sloane_functions import recur_gen2; it = recur_gen2(0,1,2,5); [next(it) for i in range(30)] # Zerinvary Lajos, Jun 25 2008

[lucas_number1(n,2,-5) for n in range(0, 26)] # Zerinvary Lajos, Apr 22 2009

From Mario Catalani (mario.catalani(AT)unito.it), Jun 14 2003: (Start)
a(2*n+1) = 5*a(n)^2 + a(n+1)^2.
6*a(2*n+1) = 5*A002533(n)^2 + A002533(n+1)^2. (End)
From Paul Barry, Sep 20 2003: (Start)
G.f.: x/(1-2*x-5*x^2).
E.g.f.: exp(x)*sinh(sqrt(6)*x)/sqrt(6).
a(n) = ((1+sqrt(6))^n - (1-sqrt(6))^n)/(2*sqrt(6)). (End)
a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*6^k. - Paul Barry, Sep 29 2004
G.f.: G(0)*x/(2*(1-x)), where G(k)= 1 + 1/(1 - x*(6*k-1)/(x*(6*k+5) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
From Peter Bala, Jul 06 2025: (Start)
For n >= 0, a(n+1) = (2^n) * Sum_{k = 0..floor(n/2)} binomial(n-k, k) * (5/4)^k.
For n >= 2, a(n) = 2^(n-1) * hypergeom([1 - (1/2)*n, 1/2 - (1/2)*n], [1 - n], -5).
Sum_{n >= 1} (-5)^n/(a(n)*a(n+1)) = -(sqrt(6) - 1).
Sum_{n >= 1} 5^n/(a(n)*a(n+2)) = 5/4; Sum_{n >= 1} 5^n/(a(n)*a(n+4)) = 755/7056.
G.f. A(x) = x*exp(Sum_{n >= 1} a(2*n)/a(n)*x^n/n) = x + 2*x^2 + 9*x^3 + 28*x^4 + .... (End)

More terms from Rick L. Shepherd, Sep 19 2004

A083099 a(n) = 2a(n-1) + 6a(n-2), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, 10, 32, 124, 440, 1624, 5888, 21520, 78368, 285856, 1041920, 3798976, 13849472, 50492800, 184082432, 671121664, 2446737920, 8920205824, 32520839168, 118562913280, 432250861568, 1575879202816, 5745263575040

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

a(n+1) = a(n) + A083098(n+1). A083098(n+1)/a(n) converges to sqrt(7).
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 7 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(7). - Cino Hilliard, Sep 25 2005
Pisano period lengths: 1, 1, 2, 1, 12, 2, 7, 1, 6, 12, 60, 2,168, 7, 12, 1,288, 6, 18, 12, ... - R. J. Mathar, Aug 10 2012
a(n) is divisible by 2^ceiling(n/2), see formula below. - Ralf Stephan, Dec 24 2013
Connect the center of a regular hexagon with side length 1 with its six vertices. a(n) is the number of paths of length n from the center to any of its vertices. Number of paths of length n from the center to itself is 6*a(n-1). - Jianing Song, Apr 20 2019

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Project Euler, Problem 752, sequence beta(n).
Index entries for linear recurrences with constant coefficients, signature (2,6).

The following sequences (and others) belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A063727, A083098, A083099, A083100, A084057.

Cf. A154245, A291008.

[n le 2 select n-1 else 2*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

A083099 := proc(n)
    option remember;
    if n <= 1 then
        n;
    else
        2*procname(n-1)+6*procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Sep 23 2016

CoefficientList[Series[x/(1-2x-6x^2), {x, 0, 25}], x] (* Adapted for offset 0 by Vincenzo Librandi, Feb 07 2014 *)
Expand[Table[((1 + Sqrt[7])^n - (1 - Sqrt[7])^n)7/(14Sqrt[7]), {n, 0, 25}]] (* Zerinvary Lajos, Mar 22 2007 *)
LinearRecurrence[{2,6}, {0,1}, 25] (* Sture Sjöstedt, Dec 06 2011 *)

a(n)=([0,1; 6,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, May 10 2016

my(x='x+O('x^30)); concat([0], Vec(x/(1-2*x-6*x^2))) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,2,-6) for n in range(0, 25)] # Zerinvary Lajos, Apr 22 2009

A083099=BinaryRecurrenceSequence(2,6,0,1)
[A083099(n) for n in range(41)] # G. C. Greubel, Jun 01 2023

G.f.: x/(1 - 2*x - 6*x^2).
From Paul Barry, Sep 29 2004: (Start)
E.g.f.: (d/dx)(exp(x)*sinh(sqrt(7)*x)/sqrt(7));
a(n-1) = Sum_{k=0..n} binomial(n, 2k+1)*7^k. (End)
Simplified formula: a(n) = ((1 + sqrt(7))^n - (1 - sqrt(7))^n)/sqrt(28). - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
G.f.: G(0)*x/(2*(1-x)), where G(k) = 1 + 1/(1 - x*(7*k-1)/(x*(7*k+6) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(2n) = 2^n * A154245(n), a(2n+1) = 2^n * (5*A154245(n) - 9*A154245(n-1)). - Ralf Stephan, Dec 24 2013
a(n) = Sum_{k=1,3,5,...<=n} binomial(n,k)*7^((k-1)/2). - Vladimir Shevelev, Feb 06 2014
a(n) = i^(n-1)*6^((n-1)/2)*ChebyshevU(n-1, -i/sqrt(6)). - G. C. Greubel, Jun 01 2023

A015519 a(n) = 2a(n-1) + 7a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, 11, 36, 149, 550, 2143, 8136, 31273, 119498, 457907, 1752300, 6709949, 25685998, 98341639, 376485264, 1441362001, 5518120850, 21125775707, 80878397364, 309637224677, 1185423230902, 4538307034543, 17374576685400

Offset: 0

Olivier Gérard

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 8 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(8). - Cino Hilliard, Sep 25 2005

Pisano period lengths: 1, 2, 8, 4, 24, 8, 3, 8, 24, 24, 15, 8, 168, 6, 24, 16, 16, 24, 120, 24, ... . - R. J. Mathar, Aug 10 2012

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7).

The following sequences (and others) belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A063727, A083098, A083099, A083100, A084057.

[ n eq 1 select 0 else n eq 2 select 1 else 2*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 23 2011

LinearRecurrence[{2,7},{0,1},30] (* Harvey P. Dale, Oct 09 2017 *)

a(n)=([0,1; 7,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, May 10 2016

[lucas_number1(n,2,-7) for n in range(0, 25)] # Zerinvary Lajos, Apr 22 2009

From Mario Catalani (mario.catalani(AT)unito.it), Apr 23 2003: (Start)
a(n) = a(n-1) + A083100(n-2), n>1.
A083100(n)/a(n+1) converges to sqrt(8). (End)
From Paul Barry, Jul 17 2003: (Start)
G.f.: x/ ( 1-2*x-7*x^2 ).
a(n) = ((1+2*sqrt(2))^n-(1-2*sqrt(2))^n)*sqrt(2)/8. (End)
E.g.f.: exp(x)*sinh(2*sqrt(2)*x)/(2*sqrt(2)). - Paul Barry, Nov 20 2003
Second binomial transform is A000129(2n)/2 (A001109). - Paul Barry, Apr 21 2004
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k)*(7/2)^k*2^(n-k-1). - Paul Barry, Jul 17 2004
a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*8^k. - Paul Barry, Sep 29 2004
G.f.: G(0)*x/(2*(1-x)), where G(k)= 1 + 1/(1 - x*(8*k-1)/(x*(8*k+7) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

A002533 a(n) = 2a(n-1) + 5a(n-2), with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 7, 19, 73, 241, 847, 2899, 10033, 34561, 119287, 411379, 1419193, 4895281, 16886527, 58249459, 200931553, 693110401, 2390878567, 8247309139, 28449011113, 98134567921, 338514191407, 1167701222419, 4027973401873, 13894452915841, 47928772841047, 165329810261299

Offset: 0

N. J. A. Sloane

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 6 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(6). - Cino Hilliard, Sep 25 2005
a(n), n>0 = term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 2,1]. - Gary W. Adamson, Aug 06 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 6 types of other natural numbers. - Milan Janjic, Aug 13 2010
Pisano period lengths: 1, 1, 1, 4, 4, 1, 24, 4, 3, 4, 120, 4, 56, 24, 4, 8, 288, 3, 18, 4, ... - R. J. Mathar, Aug 10 2012
a(k*m) is divisible by a(m) if k is odd.  - Robert Israel, May 03 2024

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
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).
A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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
Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,5).

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

[(1/2)*Floor((1+Sqrt(6))^n+(1-Sqrt(6))^n): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

[n le 2 select 1 else 2*Self(n-1) + 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

A002533:=(-1+z)/(-1+2*z+5*z**2); # conjectured by Simon Plouffe in his 1992 dissertation

f[n_] := Simplify[((1 + Sqrt[6])^n + (1 - Sqrt[6])^n)/2]; Array[f, 28, 0] (* Or *)
LinearRecurrence[{2, 5}, {1, 1}, 28] (* Or *)
Table[ MatrixPower[{{1, 2}, {3, 1}}, n][[1, 1]], {n, 0, 25}]
(* Robert G. Wilson v, Sep 18 2013 *)

a(n)=([0,1; 5,2]^n*[1;1])[1,1] \\ Charles R Greathouse IV, May 10 2016

x='x+O('x^30); Vec((1-x)/(1-2*x-5*x^2)) \\ G. C. Greubel, Jan 08 2018

[lucas_number2(n,2,-5)/2 for n in range(0, 21)] # Zerinvary Lajos, Apr 30 2009

a(n)/A002532(n), n>0, converges to sqrt(6). - Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003
From Mario Catalani (mario.catalani(AT)unito.it), May 03 2003: (Start)
G.f.: (1-x)/(1-2*x-5*x^2).
a(n) = (1/2)*((1+sqrt(6))^n + (1-sqrt(6))^n).
a(n)/A083694(n) converges to sqrt(3/2).
a(n)/A083695(n) converges to sqrt(2/3).
a(n) = a(n-1) + 3*A083694(n-1).
a(n) = a(n-1) + 2*A083695(n-1), n>0. (End)
Binomial transform of expansion of cosh(sqrt(6)*x) (A000400, with interpolated zeros). E.g.f.: exp(x)*cosh(sqrt(6)*x) - Paul Barry, May 09 2003
From Mario Catalani (mario.catalani(AT)unito.it), Jun 14 2003: (Start)
a(2*n+1) = 2*a(n)*a(n+1) - (-5)^n.
a(n)^2 - 6*A002532(n)^2 = (-5)^n. (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * 6^k. - Paul Barry, Jul 25 2004
a(n) = Sum_{k=0..n} A098158(n,k)*6^(n-k). - Philippe Deléham, Dec 26 2007
If p(1)=1, and p(I)=6, for i>1, and if A is the Hessenberg matrix of order n defined by: A(i,j) = p(j-i+1) for i<=j, A(i,j)=-1 for i=j+1, and A(i,j)=0 otherwise. Then, for n>=1, a(n) = det A. - Milan Janjic, Apr 29 2010
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(6*k-1)/(x*(6*k+5) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

A083098 a(n) = 2a(n-1) + 6a(n-2).

Original entry on oeis.org

1, 1, 8, 22, 92, 316, 1184, 4264, 15632, 56848, 207488, 756064, 2757056, 10050496, 36643328, 133589632, 487039232, 1775616256, 6473467904, 23600633344, 86042074112, 313687948288, 1143628341248, 4169384372224, 15200538791936

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

a(n+1) = a(n) + 7*A083099(n-1); a(n+1)/A083099(n) converges to sqrt(7).
Binomial transform of expansion of cosh(sqrt(7)x) (A000420 with interpolated zeros: 1, 0, 7, 0, 49, 0, 343, 0, ...).
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 7 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(7). - Cino Hilliard, Sep 25 2005
a(n) is the number of compositions of n when there are 1 type of 1 and 7 types of other natural numbers. - Milan Janjic, Aug 13 2010

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Project Euler, Problem 752, sequence alpha(n).
Index entries for linear recurrences with constant coefficients, signature (2,6).

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

CoefficientList[Series[(1+6x)/(1-2x-6x^2), {x, 0, 25}], x]
LinearRecurrence[{2, 6}, {1, 1}, 25] (* Sture Sjöstedt, Dec 06 2011 *)
a[n_] := Simplify[((1 + Sqrt[7])^n + (1 - Sqrt[7])^n)/2]; Array[a, 25, 0] (* Robert G. Wilson v, Sep 18 2013 *)

x='x+O('x^30); Vec((1-x)/(1-2*x-6*x^2)) \\ G. C. Greubel, Jan 08 2018

[lucas_number2(n,2,-6)/2 for n in range(0, 25)] # Zerinvary Lajos, Apr 30 2009

G.f.: (1-x)/(1-2*x-6*x^2).
a(n) = (1+sqrt(7))^n/2 + (1-sqrt(7))^n/2.
E.g.f.: exp(x)*cosh(sqrt(7)x).
a(n) = Sum_{k=0..n} A098158(n,k)*7^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=7, (i>1), and if A is 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) = det A. - Milan Janjic, Apr 29 2010
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(7*k-1)/(x*(7*k+6) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

A083100 a(n) = 2a(n-1) + 7a(n-2).

Original entry on oeis.org

1, 9, 25, 113, 401, 1593, 5993, 23137, 88225, 338409, 1294393, 4957649, 18976049, 72655641, 278143625, 1064876737, 4076758849, 15607654857, 59752621657, 228758827313, 875786006225, 3352883803641, 12836269650857, 49142725927201

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 23 2003

a(n) = a(n-1) + 8*A015519(n). a(n)/A015519(n+1) converges to sqrt(8).

a(n-1) is the number of compositions of n when there is 1 type of 1 and 8 types of other natural numbers. - Milan Janjic, Aug 13 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 7).

Essentially a duplicate of A084058.

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) + 7*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

CoefficientList[Series[(1 + 7 x)/(1 - 2 x - 7 x^2), {x, 0, 25}], x] (* Or *) a[n_] := Simplify[((1 + Sqrt[8])^n + (1 - Sqrt[8])^n)/2]; Array[a, 25, 0] (* Or *) LinearRecurrence[{2, 7}, {1, 1}, 28] (* Or *) Table[ MatrixPower[{{1, 2}, {4, 1}}, n][[1, 1]], {n, 0, 25}] (* Robert G. Wilson v, Sep 18 2013 *)

a(n)=([0,1; 7,2]^n*[1;9])[1,1] \\ Charles R Greathouse IV, Apr 06 2016

x='x+O('x^30); Vec((1+7*x)/(1-2*x-7*x^2)) \\ G. C. Greubel, Jan 08 2018

G.f.: (1+7*x)/(1-2*x-7*x^2).
a(n) = binomial transform of 1,8,8,64,64,512. - Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009
If p[1]=1, and p[i]=8,(i>1), and if A is 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, Apr 29 2010
G.f.: G(0)/(2*x) - 1/x, where G(k)= 1 + 1/(1 - x*(8*k-1)/(x*(8*k+7) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013

A087131 a(n) = 2^n*Lucas(n), where Lucas = A000032.

Original entry on oeis.org

2, 2, 12, 32, 112, 352, 1152, 3712, 12032, 38912, 125952, 407552, 1318912, 4268032, 13811712, 44695552, 144637952, 468058112, 1514668032, 4901568512, 15861809152, 51329892352, 166107021312, 537533612032, 1739495309312

Offset: 0

Paul Barry, Aug 16 2003

Number of ways to tile an n-bracelet with two types of colored squares and four types of colored dominoes.
Inverse binomial transform of even Lucas numbers (A014448).
From L. Edson Jeffery, Apr 25 2011: (Start)
Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,4)=
(0 0 0 0 1)
(0 0 0 2 0)
(0 0 2 0 1)
(0 2 0 2 0)
(2 0 2 0 1).
Then a(n)=(Trace(A^n)-1)/2. Also a(n)=Trace((2*A_(5,1))^n), where A_(5,1)=[(0,1); (1,1)] is also a unit-primitive matrix. (End)
Also the number of connected dominating sets in the n-sun graph for n >= 3. - Eric W. Weisstein, May 02 2017
Also the number of total dominating sets in the n-sun graph for n >= 3. - Eric W. Weisstein, Apr 27 2018

Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, identity 237, p. 132.

L. Edson Jeffery, Unit-primitive matrices
Eric Weisstein's World of Mathematics, Connected Dominating Set.
Eric Weisstein's World of Mathematics, Sun Graph.
Eric Weisstein's World of Mathematics, Total Dominating Set.
Index entries for linear recurrences with constant coefficients, signature (2,4).

First differences of A006483 and A103435.

Cf. A000032, A014334, A014335, A084057, A269992.

[2] cat [2^n*Lucas(n): n in [1..30]]; // G. C. Greubel, Dec 18 2017

Table[Tr[MatrixPower[{{2, 2}, {2, 0}}, x]], {x, 1, 20}] (* Artur Jasinski, Jan 09 2007 *)
Join[{2}, Table[2^n LucasL[n], {n, 20}]] (* Eric W. Weisstein, May 02 2017 *)
Join[{2}, 2^# LucasL[#] & [Range[20]]] (* Eric W. Weisstein, May 02 2017 *)
LinearRecurrence[{2, 4}, {2, 12}, {0, 20}] (* Eric W. Weisstein, Apr 27 2018 *)
CoefficientList[Series[(2 (-1 + x))/(-1 + 2 x + 4 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 27 2018 *)

for(n=0,30, print1(if(n==0, 2, 2^n*(fibonacci(n+1) + fibonacci(n-1))), ", ")) \\ G. C. Greubel, Dec 18 2017

first(n) = Vec(2*(1-x)/(1-2*x-4*x^2) + O(x^n)) \\ Iain Fox, Dec 19 2017

[lucas_number2(n,2,-4) for n in range(0, 25)] # Zerinvary Lajos, Apr 30 2009

a(n) = 2*A084057(n).
Recurrence: a(n) = 2a(n-1) + 4a(n-2), a(0)=2, a(1)=2.
G.f.: 2*(1-x)/(1-2*x-4*x^2).
a(n) = (1+sqrt(5))^n + (1-sqrt(5))^n.
For n>=2, a(n) = Trace of matrix [({2,2},{2,0})^n]. - Artur Jasinski, Jan 09 2007
a(n) = 2*[A063727(n)-A063727(n-1)]. - R. J. Mathar, Nov 16 2007
a(n) = (5*A052899(n)-1)/2. - L. Edson Jeffery, Apr 25 2011
a(n) = [x^n] ( 1 + x + sqrt(1 + 2*x + 5*x^2) )^n for n >= 1. - Peter Bala, Jun 23 2015
Sum_{n>=1} 1/a(n) = (1/2) * A269992. - Amiram Eldar, Nov 17 2020
From Amiram Eldar, Jan 15 2022: (Start)
a(n) == 2 (mod 10).
a(n) = 5 * A014334(n) + 2.
a(n) = 10 * A014335(n) + 2. (End)

Edited by Ralf Stephan, Feb 08 2005

A015449 Expansion of (1-4x)/(1-5x-x^2).

Original entry on oeis.org

1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686, 11913527111, 61861971241, 321223383316, 1667978887821, 8661117822421, 44973567999926, 233528957822051

Offset: 0

Olivier Gérard

Row m=5 of A135597.
Binomial transform of A152187. - Johannes W. Meijer, Aug 01 2010
For n>=1, row sums of triangle
m/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....5
.2..|..1.....5....25
.3..|..1....10....25.....125
.4..|..1....10....75.....125....625
.5..|..1....15....75.....500....625....3125
.6..|..1....15...150.....500...3125....3125...15625
.7..|..1....20...150....1250...3125...18750...15625...78125
which is triangle for numbers 5^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 12 2012
a(n+1) is (for n>=0) the number of length-n strings of 6 letters {0,1,2,3,4,5} with no two adjacent nonzero letters identical. The general case (strings of L letters) is the sequence with g.f. (1+x)/(1-(L-1)*x-x^2). - Joerg Arndt, Oct 11 2012
With offset 1, the sequence is the INVERT transform (1, 5, 5*4, 5*4^2, 5*4^3, ...); i.e., of A003947. The sequence can also be obtained by taking powers of the matrix [(1,5); (1,4)] and extracting the upper left terms. - Gary W. Adamson, Jul 31 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Joerg Arndt, Matters Computational (The Fxtbook)
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 16.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (5,1).

Cf. A084057, A108306, A164549.

a:=[1,1];; for n in [3..30] do a[n]:=5*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Oct 23 2019

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

a[0]:=1: a[1]:=1: for n from 2 to 26 do a[n]:=5*a[n-1]+a[n-2] od: seq(a[n], n=0..21); # Zerinvary Lajos, Jul 26 2006

Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{1,5},#]}]&, {1,1},40]][[1]]  (* Harvey P. Dale, Mar 23 2011 *)
LinearRecurrence[{5,1}, {1,1}, 30] (* Vincenzo Librandi, Nov 06 2012 *)
CoefficientList[Series[(1-4*x)/(1-5*x-x^2), {x,0,30}], x] (* G. C. Greubel, Dec 19 2017 *)
Sum[Fibonacci[Range[30] +k-2, 5], {k,0,1}] (* G. C. Greubel, Oct 23 2019 *)

Vec((1-4*x)/(1-5*x-x^2) +O('x^30)) \\ _G. C. Greubel, Dec 19 2017

def A015449_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x)/(1-5*x-x^2)).list()
A015449_list(30) # G. C. Greubel, Oct 23 2019

a(n) = 5*a(n-1) + a(n-2).
a(n) = Sum_{k=0..n} 4^k*A055830(n,k). - Philippe Deléham, Oct 18 2006
G.f.: (1-4*x)/(1-5*x-x^2). - Philippe Deléham, Nov 20 2008
For n >= 2, a(n) = F_n(5) + F_(n+1)(5), where F_n(x) is Fibonacci polynomial (cf. A049310): F_n(x) = Sum_{i=0..floor((n-1)/2)} C(n-i-1,i)*x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012
a(n) = Sum_{k=0..n} A046854(n-1,k)*5^k. - R. J. Mathar, Feb 10 2024

cp's OEIS Frontend

A026150 a(0) = a(1) = 1; a(n+2) = 2*a(n+1) + 2*a(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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A002532 a(n) = 2*a(n-1) + 5*a(n-2), a(0) = 0, a(1) = 1.

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A083099 a(n) = 2*a(n-1) + 6*a(n-2), a(0) = 0, a(1) = 1.

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A015519 a(n) = 2*a(n-1) + 7*a(n-2), with a(0) = 0, a(1) = 1.

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A026150 a(0) = a(1) = 1; a(n+2) = 2a(n+1) + 2a(n).

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

A002532 a(n) = 2a(n-1) + 5a(n-2), a(0) = 0, a(1) = 1.

A083099 a(n) = 2a(n-1) + 6a(n-2), a(0) = 0, a(1) = 1.

A015519 a(n) = 2a(n-1) + 7a(n-2), with a(0) = 0, a(1) = 1.

A002533 a(n) = 2a(n-1) + 5a(n-2), with a(0) = a(1) = 1.

A083098 a(n) = 2a(n-1) + 6a(n-2).

A083100 a(n) = 2a(n-1) + 7a(n-2).

A015449 Expansion of (1-4x)/(1-5x-x^2).