A015518 -id:A015518 - cp's OEIS frontend

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

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)

A009545 Expansion of e.g.f. sin(x)*exp(x).

Original entry on oeis.org

0, 1, 2, 2, 0, -4, -8, -8, 0, 16, 32, 32, 0, -64, -128, -128, 0, 256, 512, 512, 0, -1024, -2048, -2048, 0, 4096, 8192, 8192, 0, -16384, -32768, -32768, 0, 65536, 131072, 131072, 0, -262144, -524288, -524288, 0, 1048576, 2097152, 2097152, 0, -4194304, -8388608, -8388608, 0, 16777216, 33554432

Offset: 0

R. H. Hardin

Also first of the two associated sequences a(n) and b(n) built from a(0)=0 and b(0)=1 with the formulas a(n) = a(n-1) + b(n-1) and b(n) = -a(n-1) + b(n-1). The initial terms of the second sequence b(n) are 1, 1, 0, -2, -4, -4, 0, 8, 16, 16, 0, -32, -64, -64, 0, 128, 256, ... The points Mn(a(n)+b(n)*I) of the complex plane are located on the spiral logarithmic rho = 2*(1/2)^(2*theta)/Pi) and on the straight lines drawn from the origin with slopes: infinity, 1/2, 0, -1/2. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 30 2007
A000225: (1, 3, 7, 15, 31, ...) = 2^n - 1 = INVERT transform of A009545 starting (1, 2, 2, 0, -4, -8, ...). (Cf. comments in A144081). - Gary W. Adamson, Sep 10 2008
Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012
The variant 0, 1, -2, 2, 0, -4, 8, -8, 0, 16, -32, 32, 0, -64, (with different signs) is the Lucas U(-2,2) sequence. - R. J. Mathar, Jan 08 2013
(1+i)^n = A146559(n) + a(n)*i where i = sqrt(-1). - Philippe Deléham, Feb 13 2013
This is the Lucas U(2,2) sequence. - Raphie Frank, Nov 28 2015
{A146559, A009545} are the difference analogs of {cos(x),sin(x)} (cf. [Shevelev] link). - Vladimir Shevelev, Jun 08 2017

N. J. A. Sloane, Table of n, a(n) for n = 0..2000, Apr 09 2016 (first 100 terms from T. D. Noe)
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=2, q=-2.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45), lhs, m=2.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
N. J. A. Sloane, Table of n, (I-1)^n for n = 0..100
Wikipedia, Lucas sequence.
Index entries for linear recurrences with constant coefficients, signature (2,-2).
Index entries for sequences related to Chebyshev polynomials
Index entries for Lucas sequences

Cf. A009116. For minor variants of this sequence see A108520, A084102, A099087.
a(2*n) = A056594(n)*2^n, n >= 1, a(2*n+1) = A057077(n)*2^n.
This is the next term in the sequence A015518, A002605, A000129, A000079, A001477.
Cf. A000225, A144081. - Gary W. Adamson, Sep 10 2008
Cf. A146559.

I:=[0,1,2,2]; [n le 4 select I[n] else -4*Self(n-4): n in [1..60]]; // Vincenzo Librandi, Nov 29 2015

t1 := sum(n*x^n, n=0..100): F := series(t1/(1+x*t1), x, 100): for i from 0 to 50 do printf(`%d, `, coeff(F, x, i)) od: # Zerinvary Lajos, Mar 22 2009
G(x):=exp(x)*sin(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..50 ); # Zerinvary Lajos, Apr 05 2009
A009545 := n -> `if`(n<2, n, 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], 2)):
seq(simplify(A009545(n)), n=0..50); # Peter Luschny, Dec 17 2015

nn=104; Range[0,nn-1]! CoefficientList[Series[Sin[x]Exp[x], {x,0,nn}], x] (* T. D. Noe, May 26 2007 *)
Join[{a=0,b=1},Table[c=2*b-2*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
f[n_] := (1 + I)^(n - 2) + (1 - I)^(n - 2); Array[f, 51, 0] (* Robert G. Wilson v, May 30 2011 *)
LinearRecurrence[{2,-2},{0,1},110] (* Harvey P. Dale, Oct 13 2011 *)

x='x+O('x^66); Vec(serlaplace(exp(x)*sin(x))) /* Joerg Arndt, Apr 24 2011 */

x='x+O('x^100); concat(0, Vec(x/(1-2*x+2*x^2))) \\ Altug Alkan, Dec 04 2015

def A009545(n): return ((0, 1, 2, 2)[n&3]<<((n>>1)&-2))*(-1 if n&4 else 1) # Chai Wah Wu, Feb 16 2024

[lucas_number1(n,2,2) for n in range(0, 51)] # Zerinvary Lajos, Apr 23 2009

def A146559():
    x, y = 0, -1
    while True:
        yield x
        x, y = x - y, x + y
a = A146559(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

a(0)=0; a(1)=1; a(2)=2; a(3)=2; a(n) = -4*a(n-4), n>3. - Larry Reeves (larryr(AT)acm.org), Aug 24 2000
Imaginary part of (1+i)^n. - Marc LeBrun
G.f.: x/(1 - 2*x + 2*x^2).
E.g.f.: sin(x)*exp(x).
a(n) = S(n-1, sqrt(2))*(sqrt(2))^(n-1) with S(n, x)= U(n, x/2) Chebyshev's polynomials of the 2nd kind, Cf. A049310, S(-1, x) := 0.
a(n) = ((1+i)^n - (1-i)^n)/(2*i) = 2*a(n-1) - 2*a(n-2) (with a(0)=0 and a(1)=1). - Henry Bottomley, May 10 2001
a(n) = (1+i)^(n-2) + (1-i)^(n-2). - Benoit Cloitre, Oct 28 2002
a(n) = Sum_{k=0..n-1} (-1)^floor(k/2)*binomial(n-1, k). - Benoit Cloitre, Jan 31 2003
a(n) = 2^(n/2)sin(Pi*n/4). - Paul Barry, Sep 17 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k+1)*(-1)^k. - Paul Barry, Sep 20 2003
a(n+1) = Sum_{k=0..n} 2^k*A109466(n,k). - Philippe Deléham, Nov 13 2006
a(n) = 2*((1/2)^(2*theta(n)/Pi))*cos(theta(n)) where theta(4*p+1) = p*Pi + Pi/2, theta(4*p+2) = p*Pi + Pi/4, theta(4*p+3) = p*Pi - Pi/4, theta(4*p+4) = p*Pi - Pi/2, or a(0)=0, a(1)=1, a(2)=2, a(3)=2, and for n>3 a(n)=-4*a(n-4). Same formulas for the second sequence replacing cosines with sines. For example: a(0) = 0, b(0) = 1; a(1) = 0+1 = 1, b(1) = -0+1 = 1; a(2) = 1+1 = 2, b(2) = -1+1 = 0; a(3) = 2+0 = 2, b(3) = -2+0 = -2. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 30 2007
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3), n > 3, which implies the sequence is identical to its fourth differences. Binomial transform of 0, 1, 0, -1. - Paul Curtz, Dec 21 2007
Logarithm g.f. arctan(x/(1-x)) = Sum_{n>0} a(n)/n*x^n. - Vladimir Kruchinin, Aug 11 2010
a(n) = A046978(n) * A016116(n). - Paul Curtz, Apr 24 2011
E.g.f.: exp(x) * sin(x) = x + x^2/(G(0)-x); G(k) = 2k + 1 + x - x*(2k+1)/(4k+3+x+x^2*(4k+3)/( (2k+2)*(4k+5) - x^2 - x*(2k+2)*(4k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2011
a(n) = Im( (1+i)^n ) where i=sqrt(-1). - Stanislav Sykora, Jun 11 2012
G.f.: x*U(0) where U(k) = 1 + x*(k+3) - x*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012
G.f.: G(0)*x/(2*(1-x)), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
G.f.: x + x^2*W(0), where W(k) = 1 + 1/(1 - x*(k+1)/( x*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 - 2*x)/( x*(4*k+4 - 2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 06 2013
a(n) = (A^n - B^n)/(A - B), where A = 1 + i and B = 1 - i; A and B are solutions of x^2 - 2*x + 2 = 0. - Raphie Frank, Nov 28 2015
a(n) = 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], 2) for n >= 2. - Peter Luschny, Dec 17 2015
a(k+m) = a(k)*A146559(m) + a(m)*A146559(k). - Vladimir Shevelev, Jun 08 2017

Extended with signs by Olivier Gérard, Mar 15 1997
More terms from Larry Reeves (larryr(AT)acm.org), Aug 24 2000
Definition corrected by Joerg Arndt, Apr 24 2011

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

Original entry on oeis.org

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

A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 5, 7, 4, 10, 6, 12, 6, 8, 11, 16, 7, 18, 12, 12, 10, 22, 10, 21, 12, 20, 18, 28, 8, 30, 21, 20, 16, 24, 21, 36, 18, 24, 20, 40, 12, 42, 30, 28, 22, 46, 22, 43, 21, 32, 36, 52, 20, 40, 30, 36, 28, 58, 24, 60, 30, 42, 43, 48, 20, 66, 48, 44, 24, 70, 35

Offset: 1

N. J. A. Sloane, Feb 06 2012

For more information see the Comments in A061020.
a(n) is the number of integers j such that 1 <= j <= n and gcd(n,j) is a perfect square. For example, a(12) = 6 because |{1,4,5,7,8,11}|=6 and the respective GCDs with 12 are 1,4,1,1,4,1, which are squares. - Geoffrey Critzer, Feb 16 2015
If m is squarefree (A005117), then a(m) = A000010(m) where A000010 is the Euler totient function. - Michel Marcus, Nov 08 2017
Also it appears that the primorials (A002110) is the sequence of indices of minimum records for a(n)/n, and these records are A038110(n)/A060753(n). - Michel Marcus, Nov 09 2017
Also called rho(n). When rho(n) | n, then n is called k-imperfect, with k = n/rho(n), cf. A127724. - M. F. Hasler, Feb 13 2020

P. J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 25.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.

Cf. A061020, A206368.
Cf. A027748 row, A124010, A206475 (first differences).
Cf. A078429.
Cf. A127724 (k-imperfect), A127725 (2-imperfect), A127726 (3-imperfect).

a206369 n = product $
   zipWith h (a027748_row n) (map toInteger $ a124010_row n) where
           h p e = sum $ take (fromInteger e + 1) $
                         iterate ((* p) . negate) (1 - 2 * (e `mod` 2))
-- Reinhard Zumkeller, Feb 08 2012

a:= n-> mul(add(i[1]^(i[2]-j)*(-1)^j, j=0..i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 03 2017

Table[Length[Select[Range[n], IntegerQ[GCD[n, #]^(1/2)] &]], {n, 72}] (* Geoffrey Critzer, Feb 16 2015 *)
a[n_] := n*DivisorSum[n, LiouvilleLambda[#]/#&]; Array[a, 72] (* Jean-François Alcover, Dec 04 2017, after Enrique Pérez Herrero *)
f[p_,e_] := Sum[(-1)^(e-k)*p^k, {k,0,e}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jan 01 2020 *)

a(n) = sum(k=1, n, issquare(gcd(n, k)));

ak(p,e)=my(s=1); for(i=1,e, s=s*p + (-1)^i); s
a(n)=my(f=factor(n)); prod(i=1,#f~, ak(f[i,1],f[i,2])) \\ Charles R Greathouse IV, Dec 27 2016

a(n) = sumdiv(n, d, eulerphi(n/d) * issquare(d)); \\ Daniel Suteu, Jun 27 2018

apply( {A206369(n)=vecprod([f[1]^(f[2]+1)\/(f[1]+1)|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Feb 13 2020

from math import prod
from sympy import factorint
def A206369(n): return prod((lambda x:x[0]+int((x[1]<<1)>=p+1))(divmod(p**(e+1),p+1)) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 05 2024

a(n) = abs(A061020(n)).
a(n) = n*Sum_{d|n} lambda(d)/d, where lambda(n) is A008836(n). - Enrique Pérez Herrero, Sep 23 2012
Dirichlet g.f.: zeta(s - 1)*zeta(2*s)/zeta(s). - Geoffrey Critzer, Feb 25 2015
From Michel Marcus, Nov 05 2017: (Start)
a(2^n) = A001045(n+1);
a(3^n) = A015518(n+1);
a(5^n) = A015531(n+1);
a(7^n) = A015552(n+1);
a(11^n) = A015592(n+1). (End)
a(p^k) = p^k - a(p^(k - 1)) for k > 0 and prime p. - David A. Corneth, Nov 09 2017
a(n) = Sum_{d|n, d is a perfect square} phi(n/d), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 27 2018
a(p^k) = A071324(p^k), for k >= 0 and prime p. - Michel Marcus, Aug 11 2018
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 30. - Vaclav Kotesovec, Feb 07 2019
G.f.: Sum_{k>=1} lambda(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, May 23 2019
a(n) = Sum_{i=1..n} A010052(gcd(n,i)). - Ridouane Oudra, Nov 24 2019
a(p^k) = round(p^(k+1)/(p+1)). - M. F. Hasler, Feb 13 2020

A007482 a(n) is the number of subsequences of [ 1, ..., 2n ] in which each odd number has an even neighbor.

Original entry on oeis.org

1, 3, 11, 39, 139, 495, 1763, 6279, 22363, 79647, 283667, 1010295, 3598219, 12815247, 45642179, 162557031, 578955451, 2061980415, 7343852147, 26155517271, 93154256107, 331773802863, 1181629920803, 4208437368135

Offset: 0

N. J. A. Sloane

The even neighbor must differ from the odd number by exactly one.
If we defined this sequence by the recurrence (a(n) = 3*a(n-1) + 2*a(n-2)) that it satisfies, we could prefix it with an initial 0.
a(n) equals term (1,2) in M^n, M = the 3 X 3 matrix [1,1,2; 1,0,1; 2,1,1]. - Gary W. Adamson, Mar 12 2009
a(n) equals term (2,2) in M^n, M = the 3 X 3 matrix [0,1,0; 1,3,1; 0,1,0]. - Paul Barry, Sep 18 2009
From Gary W. Adamson, Aug 06 2010: (Start)
Starting with "1" = INVERT transform of A002605: (1, 2, 6, 16, 44, ...).
Example: a(3) = 39 = (16, 6, 2, 1) dot (1, 1, 3, 11) = (16 + 6 + 6 + 11). (End)
Pisano periods: 1, 1, 4, 1, 24, 4, 48, 2, 12, 24, 30, 4, 12, 48, 24, 4,272, 12, 18, 24, ... . - R. J. Mathar, Aug 10 2012
A007482 is also the number of ways of tiling a 3 X n rectangle with 1 X 1 squares, 2 X 2 squares and 2 X 1 (vertical) dominoes. - R. K. Guy, May 20 2015
With offset 1 (a(0) = 0, a(1) = 1) this is a divisibility sequence. - Michael Somos, Jun 03 2015
Number of elements of size 2^(-n) in a fractal generated by the second-order reversible cellular automaton, rule 150R (see the reference and the link). - Yuriy Sibirmovsky, Oct 04 2016
a(n) is the number of compositions (ordered partitions) of n into parts 1 (of three kinds) and 2 (of two kinds). - Joerg Arndt, Oct 05 2016
a(n) equals the number of words of length n over {0,1,2,3,4} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017
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 two '1's in their modulo 3 coordinates. a(n) is the number of cells after n iterations. Cell configuration converges to a fractal with approximate dimension 1.833. - Peter Karpov, Apr 20 2017
This is the Lucas sequence U(P=3,Q=-2), and hence for n>=0, a(n+2)/a(n+1) equals the continued fraction 3 + 2/(3 + 2/(3 + 2/(3 + ... + 2/3))) with n 2's. - Greg Dresden, Oct 06 2019

			G.f. = 1 + 3*x + 11*x^2 + 39*x^3 + 139*x^4 + 495*x^5 + 1763*x^6 + ...
From _M. F. Hasler_, Jun 16 2019: (Start)
For n = 0, (1, ..., 2n) = () is the empty sequence, which is equal to its only subsequence, which satisfies the condition voidly, whence a(0) = 1.
For n = 1, (1, ..., 2n) = (1, 2); among the four subsequences {(), (1), (2), (1,2)} only (1) does not satisfy the condition, whence a(1) = 3.
For n = 2, (1, ..., 2n) = (1, 2, 3, 4); among the sixteen subsequences {(), ..., (1,2,3,4)}, the 5 subsequences (1), (3), (1,3), (2,3,4) and (1,2,3,4) do not satisfy the condition, whence a(2) = 16 - 5 = 11.
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 439.

T. D. Noe, Table of n, a(n) for n = 0..200
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 13, 29.
Alexander Burstein and Opel Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, arXiv:2002.12189 [math.CO], 2020.
R. K. Guy and William O. J. Moser, Numbers of subsequences without isolated odd members, Fibonacci Quarterly, 34, No. 2, 152-155 (1996). Math. Rev. 97d:11017.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 442
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..8)
Yuriy Sibirmovsky, A fractal with number of elements described by a(n)
Index entries for linear recurrences with constant coefficients, signature (3,2).

Row sums of triangle A073387.
Cf. A000045, A000129, A001045, A007455, A007481, A007483, A007484, A015518, A201000 (prime subsequence), A052913 (binomial transform), A026597 (inverse binomial transform).
Cf. A206776.

a007482 n = a007482_list !! (n-1)
a007482_list = 1 : 3 : zipWith (+)
               (map (* 3) $ tail a007482_list) (map (* 2) a007482_list)
-- Reinhard Zumkeller, Oct 21 2015

I:=[1,3]; [n le 2 select I[n] else 3*Self(n-1) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

a := n -> `if`(n=0, 1, 3^n*hypergeom([(1-n)/2,-n/2], [-n], -8/9)):
seq(simplify(a(n)), n = 0..23); # Peter Luschny, Jun 28 2017

a[n_]:=(MatrixPower[{{1,4},{1,2}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{3,2},{1,3},30] (* Harvey P. Dale, May 25 2013 *)
a[ n_] := Module[ {m = n + 1, s = 1}, If[ m < 0, {m, s} = -{m, (-2)^m}]; s SeriesCoefficient[ x / (1 - 3 x - 2 x^2), {x, 0, m}]]; (* Michael Somos, Jun 03 2015 *)
a[ n_] := With[{m = n + 1}, If[ m < 0, (-2)^m a[ -m], Expand[((3 + Sqrt[17])/2)^m - ((3 - Sqrt[17])/2)^m ] / Sqrt[17]]]; (* Michael Somos, Oct 13 2016 *)

a(n) := if n=0 then 1 elseif n=1 then 3 else 3*a(n-1)+2*a(n-2);
makelist(a(n),n,0,12); /* Emanuele Munarini, Jun 28 2017 */

{a(n) = 2*imag(( (3 + quadgen(68)) / 2)^(n+1))}; /* Michael Somos, Jun 03 2015 */

[lucas_number1(n,3,-2) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

G.f.: 1/(1-3*x-2*x^2).
a(n) = 3*a(n-1) + 2*a(n-2).
a(n) = (ap^(n+1)-am^(n+1))/(ap-am), where ap = (3+sqrt(17))/2 and am = (3-sqrt(17))/2.
Let b(0) = 1, b(k) = floor(b(k-1)) + 2/b(k-1); then, for n>0, b(n) = a(n)/a(n-1). - Benoit Cloitre, Sep 09 2002
The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)2^k*3^(n-2k). - Paul Barry, Apr 23 2005
a(n) = Sum_{k=0..n} A112906(n,k). - Philippe Deléham, Nov 21 2007
a(n) = - a(-2-n) * (-2)^(n+1) for all n in Z. - Michael Somos, Jun 03 2015
If c = (3 + sqrt(17))/2, then c^n = (A206776(n) + sqrt(17)*a(n-1)) / 2. - Michael Somos, Oct 13 2016
a(n) = 3^n*hypergeom([(1-n)/2,-n/2], [-n], -8/9) for n>=1. - Peter Luschny, Jun 28 2017
a(n) = round(((sqrt(17) + 3)/2)^(n+1)/sqrt(17)). The distance of the argument from the nearest integer is about 1/2^(n+3). - M. F. Hasler, Jun 16 2019
E.g.f.: (1/17)*exp(3*x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2)). - Stefano Spezia, Oct 07 2019
a(n) = (sqrt(2)*i)^n * ChebyshevU(n, -3*i/(2*sqrt(2))). - G. C. Greubel, Dec 24 2021
G.f.: 1/(1 - 3*x - 2*x^2) = Sum_{n >= 0} x^n * Product_{k = 1..n} (k + 2*x + 2)/(1 + k*x) (a telescoping series). Cf. A015518. - Peter Bala, May 08 2024

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

A135278 Triangle read by rows, giving the numbers T(n,m) = binomial(n+1, m+1); or, Pascal's triangle A007318 with its left-hand edge removed.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 6, 15, 20, 15, 6, 1, 7, 21, 35, 35, 21, 7, 1, 8, 28, 56, 70, 56, 28, 8, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 12, 66, 220, 495, 792, 924, 792

Offset: 0

Zerinvary Lajos, Dec 02 2007

T(n,m) is the number of m-faces of a regular n-simplex.
An n-simplex is the n-dimensional analog of a triangle. Specifically, a simplex is the convex hull of a set of (n + 1) affinely independent points in some Euclidean space of dimension n or higher, i.e., a set of points such that no m-plane contains more than (m + 1) of them. Such points are said to be in general position.
Reversing the rows gives A074909, which as a linear sequence is essentially the same as this.
From Tom Copeland, Dec 07 2007: (Start)
T(n,k) * (k+1)! = A068424. The comment on permuted words in A068424 shows that T is related to combinations of letters defined by connectivity of regular polytope simplexes.
If T is the diagonally-shifted Pascal matrix, binomial(n+m, k+m), for m=1, then T is a fundamental type of matrix that is discussed in A133314 and the following hold.
The infinitesimal matrix generator is given by A132681, so T = LM(1) of A132681 with inverse LM(-1).
With a(k) = (-x)^k / k!, T * a = [ Laguerre(n,x,1) ], a vector array with index n for the Laguerre polynomials of order 1. Other formulas for the action of T are given in A132681.
T(n,k) = (1/n!) (D_x)^n (D_t)^k Gf(x,t) evaluated at x=t=0 with Gf(x,t) = exp[ t * x/(1-x) ] / (1-x)^2.
[O.g.f. for T ] = 1 / { [ 1 - t * x/(1-x) ] * (1-x)^2 }. [ O.g.f. for row sums ] = 1 / { (1-x) * (1-2x) }, giving A000225 (without a leading zero) for the row sums. Alternating sign row sums are all 1. [Sign correction noted by Vincent J. Matsko, Jul 19 2015]
O.g.f. for row polynomials = [ (1+q)**(n+1) - 1 ] / [ (1+q) -1 ] = A(1,n+1,q) on page 15 of reference on Grassmann cells in A008292. (End)
Given matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. The e.g.f. for the row polynomials of A is {(a+t) exp[(a+t)x] - a exp(a x)}/t, umbrally. - Tom Copeland, Aug 21 2008
A007318*A097806 as infinite lower triangular matrices. - Philippe Deléham, Feb 08 2009
Riordan array (1/(1-x)^2, x/(1-x)). - Philippe Deléham, Feb 22 2012
The elements of the matrix inverse are T^(-1)(n,k)=(-1)^(n+k)*T(n,k). - R. J. Mathar, Mar 12 2013
Relation to K-theory: T acting on the column vector (-0,d,-d^2,d^3,...) generates the Euler classes for a hypersurface of degree d in CP^n. Cf. Dugger p. 168 and also A104712, A111492, and A238363. - Tom Copeland, Apr 11 2014
Number of walks of length p>0 between any two distinct vertices of the complete graph K_(n+2) is W(n+2,p)=(-1)^(p-1)*Sum_{k=0..p-1} T(p-1,k)*(-n-2)^k = ((n+1)^p - (-1)^p)/(n+2) = (-1)^(p-1)*Sum_{k=0..p-1} (-n-1)^k. This is equal to (-1)^(p-1)*Phi(p,-n-1), where Phi is the cyclotomic polynomial when p is an odd prime. For K_3, see A001045; for K_4, A015518; for K_5, A015521; for K_6, A015531; for K_7, A015540. - Tom Copeland, Apr 14 2014
Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-1)^0 + A_1*(x-1)^1 + A_2*(x-1)^2 + ... + A_n*(x-1)^n. This sequence gives A_0, ..., A_n as the entries in the n-th row of this triangle, starting at n = 0. - Derek Orr, Oct 14 2014
See A074909 for associations among this array, the Bernoulli polynomials and their umbral compositional inverses, and the face polynomials of permutahedra and their duals (cf. A019538). - Tom Copeland, Nov 14 2014
From Wolfdieter Lang, Dec 10 2015: (Start)
A(r, n) = T(n+r-2, r-1) = risefac(n,r)/r! = binomial(n+r-1, r), for n >= 1 and r >= 1, gives the array with the number of independent components of a symmetric tensors of rank r (number of indices) and dimension n (indices run from 1 to n). Here risefac(n, k) is the rising factorial.
As(r, n) = T(n+1, r+1) = fallfac(n, r)/r! = binomial(n, r), r >= 1 and n >= 1 (with the triangle entries T(n, k) = 0 for n < k) gives the array with the number of independent components of an antisymmetric tensor of rank r and dimension n. Here fallfac is the falling factorial. (End)
The h-vectors associated to these f-vectors are given by A000012 regarded as a lower triangular matrix. Read as bivariate polynomials, the h-polynomials are the complete homogeneous symmetric polynomials in two variables, found in the compositional inverse of an e.g.f. for A008292, the h-vectors of the permutahedra. - Tom Copeland, Jan 10 2017
For a correlation between the states of a quantum system and the combinatorics of the n-simplex, see Boya and Dixit. - Tom Copeland, Jul 24 2017

			The triangle T(n, k) begins:
   n\k  0  1   2   3   4   5   6   7   8  9 10 11 ...
   0:   1
   1:   2  1
   2:   3  3   1
   3:   4  6   4   1
   4:   5 10  10   5   1
   5:   6 15  20  15   6   1
   6:   7 21  35  35  21   7   1
   7:   8 28  56  70  56  28   8   1
   8:   9 36  84 126 126  84  36   9   1
   9:  10 45 120 210 252 210 120  45  10  1
  10:  11 55 165 330 462 462 330 165  55 11  1
  11:  12 66 220 495 792 924 792 495 220 66 12  1
  ... reformatted by _Wolfdieter Lang_, Mar 23 2015
Production matrix begins
   2   1
  -1   1   1
   1   0   1   1
  -1   0   0   1   1
   1   0   0   0   1   1
  -1   0   0   0   0   1   1
   1   0   0   0   0   0   1   1
  -1   0   0   0   0   0   0   1   1
   1   0   0   0   0   0   0   0   1   1
- _Philippe Deléham_, Jan 29 2014
From _Wolfdieter Lang_, Nov 08 2018: (Start)
Recurrence [_Philippe Deléham_]: T(7, 3) = 2*35 + 35 - 15 - 20 = 70.
Recurrence from Riordan A- and Z-sequences: [1,1,repeat(0)] and [2, repeat(-1, +1)]: From Z: T(5, 0) = 2*5 - 10 + 10 - 5 + 1 = 6. From A: T(7, 3) = 35 + 35 = 70.
Boas-Buck column k=3 recurrence: T(7, 3) = (5/4)*(1 + 5 + 15 + 35) = 70. (End)

G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened
Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
L. Boya and K. Dixit, Geometry of density states, arXiv:808.1930 [quant-phy], 2017.
V. Buchstaber, Lectures on Toric Topology, Trends in Mathematics - New Series, Information Center for Mathematical Sciences, Vol. 10, No. 1, 2008. p. 7.
Tom Copeland, Cyclotomic polynomials in combinatorics
Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator Part VI on simplices
D. Dugger, A Geometric Introduction to K-Theory [From _Tom Copeland_, Apr 11 2014]
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 12.
Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 4.
B. Grünbaum and G. C. Shephard, Convex polytopes, Bull. London Math. Soc. (1969) 1 (3): 257-300.
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4 (Added by _Tom Copeland_, Oct 01 2015).
Justin Hughes, Representations Arising from an Action on D-neighborhoods of Cayley Graphs, 2013; slides from a talk.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Wikipedia, Simplex

Cf. A007318, A014410, A228196.
Cf. Column sequences: A000027, A000217, A000292, A000332, A000389, A000579 - A000582, A001287, A001288, A010965 - A011001, A017713 - A017764.
Cf. A000012, A008292.

for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i) od;

Flatten[Table[CoefficientList[D[1/x ((x + 1) Exp[(x + 1) z] - Exp[z]), {z, k}] /. z -> 0, x], {k, 0, 11}]]
CoefficientList[CoefficientList[Series[1/((1 - x)*(1 - x - x*y)), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Nov 22 2017 *)

for(n=0, 20, for(k=0, n, print1(1/k!*sum(i=0, n, (prod(j=0, k-1, i-j))), ", "))) \\ Derek Orr, Oct 14 2014

Trow = lambda n: sum((x+1)^j for j in (0..n)).list()
for n in (0..10): print(Trow(n)) # Peter Luschny, Jul 09 2019

T(n, k) = Sum_{j=k..n} binomial(j,k) = binomial(n+1, k+1), n >= k >= 0, else 0. (Partial sum of column k of A007318 (Pascal), or summation on the upper binomial index (Graham et al. (GKP), eq. (5.10). For the GKP reference see A007318.) - Wolfdieter Lang, Aug 22 2012
E.g.f.: 1/x*((1 + x)*exp(t*(1 + x)) - exp(t)) = 1 + (2 + x)*t + (3 + 3*x + x^2)*t^2/2! + .... The infinitesimal generator for this triangle has the sequence [2,3,4,...] on the main subdiagonal and 0's elsewhere. - Peter Bala, Jul 16 2013
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 27 2013
T(n,k) = A193862(n,k)/2^k. - Philippe Deléham, Jan 29 2014
G.f.: 1/((1-x)*(1-x-x*y)). - Philippe Deléham, Mar 13 2014
From Tom Copeland, Mar 26 2014: (Start)
[From Copeland's 2007 and 2008 comments]
A) O.g.f.: 1 / { [ 1 - t * x/(1-x) ] * (1-x)^2 } (same as Deleham's).
B) The infinitesimal generator for T is given in A132681 with m=1 (same as Bala's), which makes connections to the ubiquitous associated Laguerre polynomials of integer orders, for this case the Laguerre polynomials of order one L(n,-t,1).
C) O.g.f. of row e.g.f.s: Sum_{n>=0} L(n,-t,1) x^n = exp[t*x/(1-x)]/(1-x)^2 = 1 + (2+t)x + (3+3*t+t^2/2!)x^2 + (4+6*t+4*t^2/2!+t^3/3!)x^3+ ... .
D) E.g.f. of row o.g.f.s: ((1+t)*exp((1+t)*x)-exp(x))/t (same as Bala's).
E) E.g.f. for T(n,k)*a(n-k): {(a+t) exp[(a+t)x] - a exp(a x)}/t, umbrally. For example, for a(k)=2^k, the e.g.f. for the row o.g.f.s is {(2+t) exp[(2+t)x] - 2 exp(2x)}/t.
(End)
From Tom Copeland, Apr 28 2014: (Start)
With different indexing
A) O.g.f. by row: [(1+t)^n-1]/t.
B) O.g.f. of row o.g.f.s: {1/[1-(1+t)*x] - 1/(1-x)}/t.
C) E.g.f. of row o.g.f.s: {exp[(1+t)*x]-exp(x)}/t.
These generating functions are related to row e.g.f.s of A111492. (End)
From Tom Copeland, Sep 17 2014: (Start)
A) U(x,s,t)= x^2/[(1-t*x)(1-(s+t)x)] = Sum_{n >= 0} F(n,s,t)x^(n+2) is a generating function for bivariate row polynomials of T, e.g., F(2,s,t)= s^2 + 3s*t + 3t^2 (Buchstaber, 2008).
B) dU/dt=x^2 dU/dx with U(x,s,0)= x^2/(1-s*x) (Buchstaber, 2008).
C) U(x,s,t) = exp(t*x^2*d/dx)U(x,s,0) = U(x/(1-t*x),s,0).
D) U(x,s,t) = Sum[n >= 0, (t*x)^n L(n,-:xD:,-1)] U(x,s,0), where (:xD:)^k=x^k*(d/dx)^k and L(n,x,-1) are the Laguerre polynomials of order -1, related to normalized Lah numbers. (End)
E.g.f. satisfies the differential equation d/dt(e.g.f.(x,t)) = (x+1)*e.g.f.(x,t) + exp(t). - Vincent J. Matsko, Jul 18 2015
The e.g.f. of the Norlund generalized Bernoulli (Appell) polynomials of order m, NB(n,x;m), is given by exponentiation of the e.g.f. of the Bernoulli numbers, i.e., multiple binomial self-convolutions of the Bernoulli numbers, through the e.g.f. exp[NB(.,x;m)t] = (t/(e^t - 1))^(m+1) * e^(xt). Norlund gave the relation to the factorials (x-1)!/(x-1-n)! = (x-1) ... (x-n) = NB(n,x;n), so T(n,m) = NB(m+1,n+2;m+1)/(m+1)!. - Tom Copeland, Oct 01 2015
From Wolfdieter Lang, Nov 08 2018: (Start)
Recurrences from the A- and Z- sequences for the Riordan triangle (see the W. Lang link under A006232 with references), which are A(n) = A019590(n+1), [1, 1, repeat (0)] and Z(n) = (-1)^(n+1)*A054977(n), [2, repeat(-1, 1)]:
  T(0, 0) = 1, T(n, k) = 0 for n < k, and T(n, 0) = Sum_{j=0..n-1} Z(j)*T(n-1, j), for n >= 1, and T(n, k) = T(n-1, k-1) + T(n-1, k), for n >= m >= 1.
Boas-Buck recurrence for columns (see the Aug 10 2017 remark in A036521 also for references):
  T(n, k) = ((2 + k)/(n - k))*Sum_{j=k..n-1} T(j, k), for n >= 1, k = 0, 1, ..., n-1, and input T(n, n) = 1, for n >= 0, (the BB-sequences are alpha(n) = 2 and beta(n) = 1). (End)
T(n, k) = [x^k] Sum_{j=0..n} (x+1)^j. - Peter Luschny, Jul 09 2019

Edited by Tom Copeland and N. J. A. Sloane, Dec 11 2007

A015441 Generalized Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 7, 13, 55, 133, 463, 1261, 4039, 11605, 35839, 105469, 320503, 953317, 2876335, 8596237, 25854247, 77431669, 232557151, 697147165, 2092490071, 6275373061, 18830313487, 56482551853, 169464432775, 508359743893, 1525146340543, 4575304803901, 13726182847159

Offset: 0

Olivier Gérard

a(n) is the coefficient of x^(n-1) in the bivariate Fibonacci polynomials F(n)(x,y) = xF(n-1)(x,y) + yF(n-2)(x,y), F(0)(x,y)=0, F(1)(x,y)=1, when y=6x^2. - Mario Catalani (mario.catalani(AT)unito.it), Dec 06 2002
For n>=1: number of length-(n-1) words with letters {0,1,2,3,4,5,6,7} where no two consecutive letters are nonzero, see fxtbook link below. - Joerg Arndt, Apr 08 2011
Starting with offset 1 and convolved with (1, 3, 3, 3, ...) = A003462: (1, 4, 13, 40, ...). - Gary W. Adamson, May 28 2009
a(n) is identical to its inverse binomial transform signed. Differences: A102901. - Paul Curtz, Feb 23 2010
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, 7*a(n-2) equals the number of 7-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, 1, 1, 2, 20, 1, 6, 2, 3, 20, 5, 2, 12, 6, 20, 4, 16, 3, 18, 20, ... - R. J. Mathar, Aug 10 2012
A015441 and A015518 are the only integer sequences (from the family of homogeneous linear recurrence relation of order 2 with positive integer coefficients with initial values a(0)=0 and a(1)=1) whose ratio a(n+1)/a(n) converges to 3 as n approaches infinity. - Felix P. Muga II, Mar 14 2014
This is an autosequence of the first kind: the array of successive differences shows a main diagonal of zeros and the inverse binomial transform is identical to the sequence (with alternating signs). - Pointed out by Paul Curtz, Dec 05 2016
First two upper diagonals: A000400(n).
This is a variation on the Starhex honeycomb configuration A332243, see illustration in links. It is an alternating pattern of the 2nd iteration of the centered hexagonal numbers A003215 and centered 12-gonal 'Star' numbers A003154. - John Elias, Oct 06 2021

			G.f. = x + x^2 + 7*x^3 + 13*x^4 + 55*x^5 + 133*x^6 + 463*x^7 + 1261*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook), p. 317-318
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
John Elias, Illustration: A Starhex honeycomb variation
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
H. Li and T. MacHenry, Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences, J. Int. Seq. 16 (2013) #13.3.5, example 48.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
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.
Index entries for linear recurrences with constant coefficients, signature (1,6).

Cf. A000079, A003462, A016153, A109466, A122117. Cf. A001656, A087451. Cf. A000400.

I:=[0,1]; [n le 2 select I[n] else Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

A015441:=n->(1/5)*((3^n)-((-2)^n)); seq(A015441(n), n=0..30); # Wesley Ivan Hurt, Mar 14 2014

a[n_]:=(MatrixPower[{{1,4},{1,-2}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]], {n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{1,6},{0,1},30] (* Harvey P. Dale, Apr 26 2011 *)
CoefficientList[Series[x/((1 + 2 x) (1 - 3 x)), {x, 0, 29}], x] (* Michael De Vlieger, Dec 05 2016 *)

PARI
```
{a(n) = (3^n - (-2)^n) / 5};
```

[lucas_number1(n,1,-6) for n in range(0, 27)] # Zerinvary Lajos, Apr 22 2009

G.f.: x/((1+2*x)*(1-3*x)).
a(n) = a(n-1) + 6*a(n-2).
a(n) = (1/5)*((3^n)-((-2)^n)). - henryk.wicke(AT)stud.uni-hannover.de
E.g.f.: (exp(3*x) - exp(-2*x))/5. - Paul Barry, Apr 20 2003
a(n+1) = Sum_{k=0..ceiling(n/2)} 6^k*binomial(n-k, k). - Benoit Cloitre, Mar 06 2004
a(n) = (A000244(n) - A001045(n+1)(-1)^n - A001045(n)(-1)^n)/5. - Paul Barry, Apr 27 2004
The binomial transform of [1,1,7,13,55,133,463,...] is A122117. - Philippe Deléham, Oct 19 2006
a(n+1) = Sum_{k=0..n} A109466(n,k)*(-6)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = 3a(n-1) + (-1)^(n+1)*A000079(n-1). - Paul Curtz, Feb 23 2010
G.f.: Q(0) -1, where Q(k) = 1 + 6*x^2 + (k+2)*x - x*(k+1 + 6*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 06 2013
a(n) = (Sum_{1<=k<=n, k odd} binomial(n,k)*5^(k-1))/2^(n-1). - Vladimir Shevelev, Feb 05 2014
a(-n) = -(-1)^n * a(n) / 6^n for all n in Z. - Michael Somos, Mar 18 2014
From Peter Bala, Apr 01 2015: (Start)
Sum_{n >= 0} a(n+1)*x^n = exp( Sum_{n >= 1} A087451(n)*x^n/n ).
For k = 0, 1, 2, ... and for n >= 1, (5^k)*a(n) | a((5^k)*n).
The expansion of exp( Sum_{n >= 1} a(5*n)/(5*a(n))*x^n/n ) has integral coefficients. Cf. A001656. (End)
From  Peter Bala, Jun 27 2025: (Start)
Sum_{n >= 1} (-6)^n/(a(n)*a(n+1)) = -2, since (-6)^n/(a(n)*a(n+1)) = (-2)^n/a(n) - (-2)^(n+1)/a(n+1) for n >= 1.
The following are examples of telescoping infinite products:
Product_{n >= 0} (1 + 6^n/a(2*n+2)) = 6, since (1 + 6^(2*n-1)/a(4*n))*(1 + 6^(2*n)/a(4*n+2)) = (6 - 4^(n+1)/b(n)) / (6 - 4^n/b(n-1)), where b(n) = (2*4^n + 3*9^n)/5 = A096951(n). Similarly,
Product_{n >= 1} (1 - 6^n/a(2*n+2)) = 3/13.
Product_{n >= 0} (1 + (-6)^n/a(2*n+2)) = 6/5.
Product_{n >= 1} (1 - (-6)^n/a(2*n+2)) = 15/13.
exp( Sum_{n >= 1} a(2*n)/a(n)*x^n/n ) = Sum_{n >= 0} a(n+1)*x^n. (End)

A084057 a(n) = 2a(n-1) + 4a(n-2), a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 6, 16, 56, 176, 576, 1856, 6016, 19456, 62976, 203776, 659456, 2134016, 6905856, 22347776, 72318976, 234029056, 757334016, 2450784256, 7930904576, 25664946176, 83053510656, 268766806016, 869747654656, 2814562533376, 9108115685376, 29474481504256

Offset: 0

Paul Barry, May 10 2003

Inverse binomial transform of A001077. Binomial transform of expansion of cosh(sqrt(5)*x) (1,0,5,0,25,...).
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
Numerators of fractions in the approximation of the square root of 5 satisfying: a(n) = (a(n-1)+c)/(a(n-1)+1), with c=5 and a(1)=1. For denominators see A063727. - Mark Dols, Jul 24 2009
Equals right border of triangle A143969. (1, 6, 16, 56, ...) = row sums of triangle A143969 and INVERT transform of (1, 5, 5, 5, ...). - Gary W. Adamson, Sep 06 2008
a(n) is the number of compositions of n when there are 1 type of 1 and 5 types of other natural numbers. - Milan Janjic, Aug 13 2010
From Gary W. Adamson, Jul 30 2016: (Start)
The sequence is case N=1 in an infinite set obtained by taking powers of the 2 X 2 matrix M = [(1,5); (1,N)], then extracting the upper left terms. The infinite set begins:
N=1 (A084057):  1, 6, 16,  56,  176,   576,   1856, ...
N=2 (A108306):  1, 6, 21,  81,  306,  1161,   4401, ...
N=3 (A164549):  1, 6, 26, 116,  516,  2296,  10216, ...
N=4 (A015449):  1, 6, 31, 161,  836,  4341,  22541, ...
N=5 (A000400):  1, 6, 36, 216, 1296,  7776,  46656, ...
N=6 (A049685):  1, 6, 41, 281, 1926, 13201,  90481, ...
N=7 (.......):  1, 6, 46, 356, 2756, 21336, 222712, ...
...
Sequences in the above set can be obtained by taking INVERT transforms of the following:
N=1 INVERT transform of (1, 5,  5,  5,   5,    5, ...
N=2 ..."......"......". (1, 5, 10, 20,  40,   80, ...
N=3 ..."......"......". (1, 5, 15, 45, 135,  405, ...
N=4 ..."......"......". (1, 5, 20, 80, 320, 1280, ...
...
with the pattern (1, 5, N*5, (N^2)*5, (N^3)*5, ...
It appears that the sequence generated from powers (n>0) of the matrix P = [(1,a); (1,b)], (a,b > 0), then extracting the upper left terms, is equal to the INVERT transform of the sequence starting: (1, a, b*a, (b^2)*a, (b^3)*a, ...). (End)

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 (2,4).

Cf. A046717, A002533, A098158, A143969, A269992.
a(n) = A087131(n)/2.
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. A108306, A164549, A015449, A000400, A049685

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jul 31 2016

f[n_] := Simplify[((1 + Sqrt[5])^n + (1 - Sqrt[5])^n)/2]; Array[f, 28, 0] (* Or *)
LinearRecurrence[{2, 4}, {1, 1}, 28] (* Robert G. Wilson v, Sep 18 2013 *)
RecurrenceTable[{a[1] == 1, a[2] == 1, a[n] == 2 a[n-1] + 4 a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Jul 31 2016 *)
Table[2^(n-1) LucasL[n], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 19 2016 *)

lucas(n)=fibonacci(n-1)+fibonacci(n+1)
a(n)=lucas(n)/2*2^n \\ Charles R Greathouse IV, Sep 18 2013

from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,1,2,4, lambda n: 0); [next(it) for i in range(1,26)] # Zerinvary Lajos, Jul 09 2008

[lucas_number2(n,2,-4)/2 for n in range(0, 26)] # Zerinvary Lajos, Apr 30 2009

a(n) = ((1+sqrt(5))^n + (1-sqrt(5))^n)/2.
G.f.: (1-x) / (1-2*x-4*x^2).
E.g.f.: exp(x) * cosh(sqrt(5)*x).
a(2n+1) = 2*a(n)*a(n+1) - (-4)^n. - Mario Catalani (mario.catalani(AT)unito.it), Jun 13 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*5^k . - Paul Barry, Jul 25 2004
a(n) = Sum_{k=0..n} A098158(n,k)*5^(n-k). - Philippe Deléham, Dec 26 2007
a(n) = 2^(n-1)*A000032(n). - Mark Dols, Jul 24 2009
If p(1)=1, and p(i)=5 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, (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*(5*k-1)/(x*(5*k+4) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = A063727(n) - A063272(n-1). - R. J. Mathar, Jun 06 2019
a(n) = 1 + 5*A014335(n). - R. J. Mathar, Jun 06 2019
Sum_{n>=1} 1/a(n) = A269992. - Amiram Eldar, Feb 01 2021

A210246 Polylogarithm li(-n,-1/3) multiplied by (4^(n+1))/3.

Original entry on oeis.org

1, -1, -2, 2, 40, 104, -1232, -13168, 16000, 1483904, 9695488, -151161088, -3287997440, 146760704, 866038110208, 10263094740992, -169941494497280, -6324725967978496, -15215847186563072, 2895126258819203072, 54295929047166484480

Offset: 0

Stanislav Sykora, Mar 19 2012

sign

Given n, consider the series s(n) = li(-n,-1/3) = SUM((-1)^k (k^n)/3^k) for k=0,1,2,... . Then a(n)=s(n)*(4^(n+1))/3. For more details, see A212846.

			a(5) = polylog(-5,-1/3)*4^6/3 = 104.

Seiichi Manyama, Table of n, a(n) for n = 0..436 (terms 0..100 from Stanislav Sykora)
OEIS-Wiki, Eulerian polynomials
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), Stan's Library Vol. I, April 2006, updated March 2012. See Eq.(29).
Eric Weisstein's World of Mathematics, Polylogarithm

Similar to A210244. Cf. A210247 (sign changes).
Cf. A212846 (li(-n,-1/2)), A212847 (li(-n,-2/3)).
CF. A213127 through A213157.

seq(add((-1)^(n-k)*combinat[eulerian1](n,k)*3^k,k=0..n),n=0..20); # Peter Luschny, Apr 21 2013

Table[PolyLog[-n, -1/3] (4^(n+1))/3, {n, 30}] (* T. D. Noe, Mar 23 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], PolyLog[ -n, -1/3] 4^(n + 1) / 3]; (* Michael Somos, Nov 01 2014 *)

/* See in A212846, run limnpq(nmax,1,3) */

x='x+O('x^66); Vec(serlaplace( 4/(3+exp(4*x)) )) \\ Joerg Arndt, Apr 21 2013

a(n) = sum(k=0, n, k!*(-1)^k*4^(n-k)*stirling(n, k, 2)); \\ Seiichi Manyama, Mar 13 2022

Recurrence: s(n+1)=(-1/4)*SUM(C(n+1,i)*s(i)), where i=0,1,2,...,n, C(n,m) are binomial coefficients, and the starting value is s(0)=SUM((-1/3)^k)=3/4.
From Peter Bala, Mar 12 2013: (Start)
E.g.f.: A(x) = 4/(3 + exp(4*x)) = 1 - x - 2*x^2/2! + 2*x^3/3! + 40*x^4/4! + ....
The compositional inverse (A(-x) - 1)^(-1) = x + 2*x^2/2 + 7*x^3/3 + 20*x^4/4 + 61*x^5/5 + ... is the logarithmic generating function for A015518.
Recurrence equation: a(n+1) = -4*a(n) + 3*sum {k = 0..n} binomial(n,k)*a(k)*a(n-k), with a(0) = 1.
(End)
G.f.: 1 + x/Q(0), where Q(k) = 2*x*(k+1) - 1 + 3*x^2*(k+1)*(k+2)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Sep 22 2013
G.f.: 1/Q(0), where Q(k) = 1 + x*(k+1)/( 1 - 3*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 17 2013
E.g.f.: 2 - W(0), where W(k) = 1 + x/( 4*k+1 - x/( 1 + 4*x/( 4*k+3 - 4*x/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2014
a(n) = Sum_{k=0..n} k! * (-1)^k * 4^(n-k) * Stirling2(n,k). - Seiichi Manyama, Mar 13 2022
a(n) ~ n! * cos((n+1)*arctan(Pi/log(3))) * 2^(2*n + 3) / (3 * (Pi^2 + log(3)^2)^((n+1)/2)). - Vaclav Kotesovec, May 17 2022

cp's OEIS Frontend

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

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A009545 Expansion of e.g.f. sin(x)*exp(x).

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

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A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity.

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A007482 a(n) is the number of subsequences of [ 1, ..., 2n ] in which each odd number has an even neighbor.

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

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.

A084057 a(n) = 2a(n-1) + 4a(n-2), a(0)=1, a(1)=1.