A049310 -id:A049310 - cp's OEIS frontend

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

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

A051286 Whitney number of level n of the lattice of the ideals of the fence of order 2n.

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 63, 153, 376, 931, 2317, 5794, 14545, 36631, 92512, 234205, 594169, 1510192, 3844787, 9802895, 25027296, 63972861, 163701327, 419316330, 1075049011, 2758543201, 7083830648, 18204064403, 46812088751, 120452857976

Offset: 0

Emanuele Munarini

A Chebyshev transform of the central trinomial numbers A002426: image of 1/sqrt(1-2x-3x^2) under the mapping that takes g(x) to (1/(1+x^2))*g(x/(1+x^2)). - Paul Barry, Jan 31 2005
a(n) has same parity as Fibonacci(n+1) = A000045(n+1); see A107597. - Paul D. Hanna, May 22 2005
This is the second kind of Whitney numbers, which count elements, not to be confused with the first kind, which sum Mobius functions. - Thomas Zaslavsky, May 07 2008
From Paul Barry, Mar 31 2010: (Start)
Apply the Riordan array (1/(1-x+x^2),x/(1-x+x^2)) to the aerated central binomial coefficients with g.f. 1/sqrt(1-4x^2).
Hankel transform is A174882. (End)
a(n) is the number of lattice paths in L[n]. The members of L[n] are lattice paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step h with weight 1, an (1,0)-step H with weight 2, a (1,1)-step U with weight 2, and a (1,-1)-step D with weight 1. The weight of a path is the sum of the weights of its steps. Example: a(3)=5 because we have hhh, hH, Hh, UD, and DU; a(4)=11 because we have hhhh, hhH, hHh, Hhh, HH, hUD, UhD, UDh, hDU, DhU, and DUh (see the Bona-Knopfmacher reference).
Apparently the number of peakless grand Motzkin paths of length n. - David Scambler, Jul 04 2013
A bijection between L[n] (as defined above) and peakless grand Motzkin paths of length n is now given in arXiv:2002.12874. - Sergi Elizalde, Jul 14 2021
a(n) is also the number of unimodal bargraphs with a centered maximum (i.e., whose column heights are weakly increasing in the left half and weakly decreasing in the right half) and semiperimeter n+1. - Sergi Elizalde, Jul 14 2021
Diagonal of the rational function 1 / ((1 - x)*(1 - y) - (x*y)^2). - Ilya Gutkovskiy, Apr 23 2025
a(n) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,2}, and no nodes have the same weight as their parent node. - John Tyler Rascoe, Jun 10 2025

			a(3) = 5 because the ideals of size 3 of the fence F(6) = { x1 < x2 > x3 < x4 > x5 < x6 } are x1*x3*x5, x1*x2*x3, x3*x4*x5, x1*x5*x6, x3*x5*x6.

Vincenzo Librandi and Alois P. Heinz, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Axel Bacher, Cyril Banderier and Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects, in International Conference on Language and Automata Theory and Applications, S. Klein, C. Martín-Vide, D. Shapira (eds), Springer, Cham, pp 195-206, 2018.
Andrei Asinowski, Axel Bacher, Cyril Banderier and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Cyril Banderier and Paweł Hitczenko, Enumeration and asymptotics of restricted compositions having the same number of parts, Disc. Appl. Math. 160 (18) (2012) 2542-2554. See Puzzle 3.1.
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Grand Dyck paths with air pockets, arXiv:2211.04914 [math.CO], 2022.
Jean-Luc Baril and José L. Ramírez, Fibonacci and Catalan paths in a wall, 2023.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Miklós Bóna and Arnold Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Steffen Eger, The Combinatorics of Weighted Vector Compositions, arXiv:1704.04964 [math.CO], 2017.
Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2021.
Edyta Hetmaniok, Barbara Smoleń and Roman Wituła, The Stirling triangles, Proceedings of the Symposium for Young Scientists in Technology, Engineering and Mathematics (SYSTEM 2017), Kaunas, Lithuania, April 28, 2017, p. 35-41.
Ivo L. Hofacker, Christian M. Reidys, and Peter F. Stadler, Symmetric circular matchings and RNA folding. Discr. Math., 312:100-112, 2012. See Prop. 5, C_l^{1}(z).
Emanuele Munarini and Norma Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
Jesús Sistos Barrón and Hua Wang, Balanced n-color compositions, Integers (2024) Vol. 24A, Art. No. A2. See pp. 4, 5, 8, 14.

Cf. main diagonal of A125250, column k=2 of A384747.

Cf. A051291, A051292, A078698, A107597, A185828 (log), A174882 (Hankel transf.).

seq( sum('binomial(i-k,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1$2, 2, 5][n+1],
     ((2*n-1)*a(n-1)+(n-1)*a(n-2)+(2*n-3)*a(n-3)-(n-2)*a(n-4))/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 11 2016

Table[Sum[Binomial[n-k,k]^2,{k,0,Floor[n/2]}],{n,0,40}] (* Emanuele Munarini, Mar 01 2011; corrected by Harvey P. Dale, Sep 12 2012 *)
CoefficientList[Series[1/Sqrt[1-2*x-x^2-2*x^3+x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 05 2013 *)
a[n_] := HypergeometricPFQ[ {(1-n)/2, (1-n)/2, -n/2, -n/2}, {1, -n, -n}, 16]; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Feb 26 2013 *)

makelist(sum(binomial(n-k,k)^2,k,0,floor(n/2)),n,0,40);  /* Emanuele Munarini, Mar 01 2011 */

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

a(n)=sum(k=0,n,binomial(n-k,k)^2) /* Paul D. Hanna */

{a(n)=polcoeff( exp(sum(m=1,n, sum(k=0,m, binomial(2*m,2*k)*x^k) *x^m/m) +x*O(x^n)), n)}  /* Paul D. Hanna, Mar 18 2011 */

{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n, x^m*sum(k=0, n, binomial(m+k, k)^2*x^k) * (1-x)^(2*m+1) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n\2, x^(2*m) * sum(k=0, n, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n\2, x^(2*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

from sympy import binomial
def a(n): return sum(binomial(n - k, k)**2 for k in range(n//2 + 1))
print([a(n) for n in range(31)]) # Indranil Ghosh, Apr 18 2017

G.f.: 1/sqrt(1 - 2*x - x^2 - 2*x^3 + x^4).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A002426(n-2k). - Paul Barry, Jan 31 2005
From Paul D. Hanna, May 22 2005: (Start)
a(n) = Sum_{k=0..n} C(n-k, k)^2.
Limit_{n->oo} a(n+1)/a(n) = (sqrt(5)+3)/2.
G.f.: 1/sqrt((1+x+x^2)*(1-3*x+x^2)). (End)
a(n) = Sum_{k=0..n} A049310(n, k)^2. - Philippe Deléham, Nov 21 2005
a(n) = Sum_{k=0..n} (C(k,k/2)*(1+(-1)^k)/2) * Sum_{j=0..n} (-1)^((n-j)/2)*C((n+j)/2,j)*((1+(-1)^(n-j))/2)*C(j,k). - Paul Barry, Mar 31 2010
G.f.: exp( Sum_{n>=1} (x^n/n)*Sum_{k=0..n} C(2n,2k)*x^k ). - Paul D. Hanna, Mar 18 2011
Logarithmic derivative equals A185828. - Paul D. Hanna, Mar 18 2011
D-finite with recurrence: n*a(n) - (2*n-1)*a(n-1) - (n-1)*a(n-2) - (2*n-3)*a(n-3) + (n-2)*a(n-4) = 0. - R. J. Mathar, Dec 17 2011
The g.f. A(x) satisfies the differential equation (1-2*x-x^2-2*x^3+x^4)*A'(x) = (1+x+3*x^2-2*x^3)*A(x), from which the recurrence conjectured by Mathar follows. - Emanuele Munarini, Dec 18 2017
a(n) ~ phi^(2*n + 2) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 05 2013, simplified Dec 18 2017
From Paul D. Hanna, Sep 05 2014: (Start)
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^k.
G.f.: Sum_{n>=0} x^n *[Sum_{k>=0} C(n+k,k)^2 * x^k] * (1-x)^(2*n+1).
G.f.: Sum_{n>=0} x^(2*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(2*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
(End)

A037027 Skew Fibonacci-Pascal triangle read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 5, 10, 9, 4, 1, 8, 20, 22, 14, 5, 1, 13, 38, 51, 40, 20, 6, 1, 21, 71, 111, 105, 65, 27, 7, 1, 34, 130, 233, 256, 190, 98, 35, 8, 1, 55, 235, 474, 594, 511, 315, 140, 44, 9, 1, 89, 420, 942, 1324, 1295, 924, 490, 192, 54, 10, 1, 144, 744, 1836

Offset: 0

Floor van Lamoen, Jan 01 1999

T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (0,1), (1,0), (2,0). - Joerg Arndt, Jun 30 2011
T(n,k) is the number of lattice paths of length n, starting from the origin and ending at (n,k), using horizontal steps H=(1,0), up steps U=(1,1) and down steps D=(1,-1), never containing UUU, DD, HD. For instance, for n=4 and k=2, we have the paths; HHUU, HUHU, HUUH, UHHU, UHUH, UUHH, UUDU, UDUU, UUUD. - Emanuele Munarini, Mar 15 2011
Row sums form Pell numbers A000129, T(n,0) forms Fibonacci numbers A000045, T(n,1) forms A001629. T(n+k,n-k) is polynomial sequence of degree k.
T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, etc.).
As a Riordan array, this is (1/(1-x-x^2),x/(1-x-x^2)). An interesting factorization is (1/(1-x^2),x/(1-x^2))*(1/(1-x),x/(1-x)) [abs(A049310) times A007318]. Diagonal sums are the Jacobsthal numbers A001045(n+1). - Paul Barry, Jul 28 2005
T(n,k) = T'(n+1,k+1), T' given by [0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 19 2005
Equals A049310 * A007318 as infinite lower triangular matrices. - Gary W. Adamson, Oct 28 2007
This triangle may also be obtained from the coefficients of the Morgan-Voyce polynomials defined by: Mv(x, n) = (x + 1)*Mv(x, n - 1) + Mv(x, n - 2). - Roger L. Bagula, Apr 09 2008
Row sums are A000129. - Roger L. Bagula, Apr 09 2008
Absolute value of coefficients of the characteristic polynomial of tridiagonal matrices with 1's along the main diagonal, and i's along the superdiagonal and the subdiagonal (where i=sqrt(-1), see Mathematica program). - John M. Campbell, Aug 23 2011
A037027 is jointly generated with A122075 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x). See the Mathematica section at A122075. - Clark Kimberling, Mar 05 2012
For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 18 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013
Row n, for n>=0, shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^n which is the denominator of the n-th convergent of the continued fraction [x+1, x+1, x+1, ...]; see A230000. - Clark Kimberling, Nov 13 2013
T(n,k) is the number of ternary words of length n having k letters 2 and avoiding a runs of odd length for the letter 0. - Milan Janjic, Jan 14 2017
Let T(m, n, k) be an m-bonacci Pascal's triangle, where T(m, n, 0) gives the values of F(m, n), the n-th m-bonacci number, and T(m, n, k) gives the values for the k-th convolution of F(m, n). Then the classic Pascal triangle is T(1, n, k) and this sequence is T(2, n, k). T(m, n, k) is the number of compositions of n using only the positive integers 1, 1' and 2 through m, with the part 1' used exactly k times. G.f. for k-th column of T(m, n, k): x/(1 - x - x^2 - ... - x^m)^k. The row sum for T(m, n, k) is the number of compositions of n using only the positive integers 1, 1' and 2 through m. G.f. for row sum of T(m, n, k): 1/(1 - 2x - x^2 - ... - x^m). - Gregory L. Simay, Jul 24 2021

			Ratio of row polynomials R(3)/R(2) = (3 + 5*x + 3*x^2 + x^3)/(2 + 2*x + x^2) = [1+x; 1+x, 1+x].
Triangle begins:
                                 1;
                              1,    1;
                           2,    2,    1;
                        3,    5,    3,    1;
                     5,   10,    9,    4,    1;
                  8,   20,   22,   14,    5,    1;
              13,   38,   51,   40,   20,    6,    1;
           21,   71,  111,  105,   65,   27,    7,    1;
        34,  130,  233,  256,  190,   98,   35,    8,    1;
     55,  235,  474,  594,  511,  315,  140,   44,    9,    1;
  89,  420,  942, 1324, 1295,  924,  490,  192,   54,   10,    1;

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Harlan J. Brothers, Pascal's Prism: Supplementary Material
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
T. Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.
P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, Nov. 2005, pp. 359-370.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials.

A038112(n) = T(2n, n). A038137 is reflected version. Maximal row entries: A038149.
Diagonal differences are in A055830. Vertical sums are in A091186.
Cf. A007318, A049310, A000129, A155161, A122542, A059283, A228196, A228576.
Some other Fibonacci-Pascal triangles: A027926, A036355, A074829, A105809, A109906, A111006, A114197, A162741, A228074.

a037027 n k = a037027_tabl !! n !! k
a037027_row n = a037027_tabl !! n
a037027_tabl = [1] : [1,1] : f [1] [1,1] where
   f xs ys = ys' : f ys ys' where
     ys' = zipWith3 (\u v w -> u + v + w) (ys ++ [0]) (xs ++ [0,0]) ([0] ++ ys)
-- Reinhard Zumkeller, Jul 07 2012

T := (n,k) -> `if`(n=0,1,binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -4)): seq(seq(simplify(T(n,k)), k=0..n), n=0..10); # Peter Luschny, Apr 25 2016
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> combinat:-fibonacci(n)); # Peter Luschny, Oct 07 2022

Mv[x, -1] = 0; Mv[x, 0] = 1; Mv[x, 1] = 1 + x; Mv[x_, n_] := Mv[x, n] = ExpandAll[(x + 1)*Mv[x, n - 1] + Mv[x, n - 2]]; Table[ CoefficientList[ Mv[x, n], x], {n, 0, 10}] // Flatten (* Roger L. Bagula, Apr 09 2008 *)
Abs[Flatten[Table[CoefficientList[CharacteristicPolynomial[Array[KroneckerDelta[#1,#2]+KroneckerDelta[#1,#2+1]*I+KroneckerDelta[#1,#2-1]*I&,{n,n}],x],x],{n,1,20}]]] (* John M. Campbell, Aug 23 2011 *)
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[(k-n)/2, (k-n+1)/2, -n, -4];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 16 2019, after Peter Luschny *)

{T(n, k) = if( k<0 || k>n, 0, if( n==0 && k==0, 1, T(n-1, k) + T(n-1, k-1) + T(n-2, k)))}; /* Michael Somos, Sep 29 2003 */

PARI
```
T(n,k)=if(nPaul D. Hanna, Feb 27 2004
```

T(n, m) = T'(n-1, m) + T'(n-2, m) + T'(n-1, m-1), where T'(n, m) = T(n, m) for n >= 0 and 0< = m <= n and T'(n, m) = 0 otherwise.
G.f.: 1/(1 - y - y*z - y^2).
G.f. for k-th column: x/(1-x-x^2)^k.
T(n, m) = Sum_{k=0..n-m} binomial(m+k, m)*binomial(k, n-k-m), n >= m >= 0, otherwise 0. - Wolfdieter Lang, Jun 17 2002
T(n, m) = ((n-m+1)*T(n, m-1) + 2*(n+m)*T(n-1, m-1))/(5*m), n >= m >= 1; T(n, 0)= A000045(n+1); T(n, m)= 0 if n < m. - Wolfdieter Lang, Apr 12 2000
Chebyshev coefficient triangle (abs(A049310)) times Pascal's triangle (A007318) as product of lower triangular matrices. T(n, k) = Sum_{j=0..n} binomial((n+j)/2, j)*(1+(-1)^(n+j))*binomial(j, k)/2. - Paul Barry, Dec 22 2004
Let R(n) = n-th row polynomial in x, with R(0)=1, then R(n+1)/R(n) equals the continued fraction [1+x;1+x, ...(1+x) occurring (n+1) times ..., 1+x] for n >= 0. - Paul D. Hanna, Feb 27 2004
T(n,k) = Sum_{j=0..n} binomial(n-j,j)*binomial(n-2*j,k); in Egorychev notation, T(n,k) = res_w(1-w-w^2)^(-k-1)*w^(-n+k+1). - Paul Barry, Sep 13 2006
Sum_{k=0..n} T(n,k)*x^k = A000045(n+1), A000129(n+1), A006190(n+1), A001076(n+1), A052918(n), A005668(n+1), A054413(n), A041025(n), A099371(n+1), A041041(n), A049666(n+1), A041061(n), A140455(n+1), A041085(n), A154597(n+1), A041113(n) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 respectively. - Philippe Deléham, Nov 29 2009
T((m+1)*n+r-1, m*n+r-1)*r/(m*n+r) = Sum_{k=1..n} k/n*T((m+1)*n-k-1, m*n-1)*(r+k,r), n >= m > 1.
T(n-1,m-1) = (m/n)*Sum_{k=1..n-m+1} k*A000045(k)*T(n-k-1,m-2), n >= m > 1. - Vladimir Kruchinin, Mar 17 2011
T(n,k) = binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -4) for n >= 1. - Peter Luschny, Apr 25 2016

Examples from Paul D. Hanna, Feb 27 2004

A057078 Periodic sequence 1,0,-1,...; expansion of (1+x)/(1+x+x^2).

Original entry on oeis.org

1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1

Offset: 0

Wolfdieter Lang, Aug 04 2000

Partial sums of signed sequence is shifted unsigned one: |a(n+2)| = A011655(n+1).
With interpolated zeros, a(n) = sin(5*Pi*n/6 + Pi/3)/sqrt(3) + cos(Pi*n/6 + Pi/6)/sqrt(3); this gives the diagonal sums of the Riordan array (1-x^2, x(1-x^2)). - Paul Barry, Feb 02 2005
From Tom Copeland, Nov 02 2014: (Start)
With a shift and a sign change the o.g.f. of this array becomes the compositional inverse of the shifted Motzkin or Riordan numbers A005043,
(x - x^2) / (1 - x + x^2) = x*(1-x) / (1 - x*(1-x)) = x*(1-x) + [x*(1-x)]^2 + ... . Expanding each term of this series and arranging like powers of x in columns gives skewed rows of the Pascal triangle and reading along the columns gives (mod-signs and indexing) A011973, A169803, and A115139 (see also A091867, A092865, A098925, and A102426 for these term-by-term expansions and A030528). (End)

			G.f. = 1 - x^2 + x^3 - x^5 + x^6 - x^8 + x^9 - x^11 + x^12 - x^14 + x^15 + ...

Winston de Greef, Table of n, a(n) for n = 0..10000
Ralph E. Griswold, Shaft Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Cf. A049310, A010892, A011655.
A049347(n) = a(-n).
Cf. A005043, A011973, A169803, A115139, A091867, A092865, A098925, A102426, A030528.
Cf. A002264, A010060, A010872, A049347, A061347, A364374.

a057078 = (1 -) . (`mod` 3)  -- Reinhard Zumkeller, Mar 22 2013

A057078:=n->1-(n mod 3); seq(A057078(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

a[n_] := {1, 0, -1}[[Mod[n, 3] + 1]] (* Jean-François Alcover, Jul 05 2013 *)
CoefficientList[Series[(1 + x) / (1 + x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 03 2014 *)
LinearRecurrence[{-1, -1},{1, 0},90] (* Ray Chandler, Sep 15 2015 *)

{a(n) = [1, 0, -1][n%3 + 1]}; /* Michael Somos, Oct 15 2008 */

def A057078():
    x, y = -1, 0
    while True:
        yield -x
        x, y = y, -x -y
a = A057078(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

a(n) = S(n, -1) + S(n - 1, -1) = S(2*n, 1); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, -1) = A049347(n). S(n, 1) = A010892(n).
From Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003: (Start)
a(n) = (1/2)*((-1)^floor(2*n/3) + (-1)^floor((2*n+1)/3)).
a(n) = -a(n-1) - a(n-2).
a(n) = A061347(n) - A049347(n+2). (End)
a(n) = Sum_{k=0..n} binomial(n+k, 2k)*(-1)^(n-k) = Sum_{k=0..floor((n+1)/2)} binomial(n+1-k, k)*(-1)^(n-k). - Mario Catalani (mario.catalani(AT)unito.it), Aug 20 2003
Binomial transform is A010892. a(n) = 2*sqrt(3)*sin(2*Pi*n/3 + Pi/3)/3. - Paul Barry, Sep 13 2003
a(n) = cos(2*Pi*n/3) + sin(2*Pi*n/3)/sqrt(3). - Paul Barry, Oct 27 2004
a(n) = Sum_{k=0..n} (-1)^A010060(2n-2k)*(binomial(2n-k, k) mod 2). - Paul Barry, Dec 11 2004
a(n) = (4/3)*(|sin(Pi*(n-2)/3)| - |sin(Pi*n/3)|)*|sin(Pi*(n-1)/3)|. - Hieronymus Fischer, Jun 27 2007
a(n) = 1 - (n mod 3) = 1 + 3*floor(n/3) - n. - Hieronymus Fischer, Jun 27 2007
a(n) = 1 - A010872(n) = 1 + 3*A002264(n) - n. - Hieronymus Fischer, Jun 27 2007
Euler transform of length 3 sequence [0, -1, 1]. - Michael Somos, Oct 15 2008
a(n) = a(n-1)^2 - a(n-2)^2 with a(0) = 1, a(1) = 0. - Francesco Daddi, Aug 02 2011
a(n) = A049347(n) + A049347(n-1). - R. J. Mathar, Jun 26 2013
E.g.f.: exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, May 16 2023
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Feb 20 2024
From Peter Bala, Sep 08 2024: (Start)
G.f. A(x) satisfies A(x) = (1 + x)*(1 - x*A(x)).
1/x * series_reversion(x/A(x)) = the g.f of A364374. (End)

A001079 a(n) = 10*a(n-1) - a(n-2); a(0) = 1, a(1) = 5.

Original entry on oeis.org

1, 5, 49, 485, 4801, 47525, 470449, 4656965, 46099201, 456335045, 4517251249, 44716177445, 442644523201, 4381729054565, 43374646022449, 429364731169925, 4250272665676801, 42073361925598085, 416483346590304049

Offset: 0

N. J. A. Sloane

Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(6). - Benoit Cloitre, Feb 14 2004
Appears to give all solutions >1 to the equation x^2=ceiling(x*r*floor(x/r)) where r=sqrt(6). - Benoit Cloitre, Feb 24 2004
a(n) and b(n) (A004189) are the nonnegative proper solutions to the Pell equation a(n)^2 - 6*(2*b(n))^2 = +1, n >= 0. The formula given below by Gregory V. Richardson follows. - Wolfdieter Lang, Jun 26 2013
a(n) are the integer square roots of (A032528 + 1). They are also the values of m where (A032528(m) - 1) has integer square roots. See A122653 for the integer square roots of (A032528 - 1), and see A122652 for the values of m where (A032528(m) + 1) has integer square roots. - Richard R. Forberg, Aug 05 2013
a(n) are also the values of m where floor(2m^2/3) has integer square roots, excluding m = 0. The corresponding integer square roots are given by A122652(n). - Richard R. Forberg, Nov 21 2013
Except for the first term, positive values of x (or y) satisfying x^2 - 10xy + y^2 + 24 = 0. - Colin Barker, Feb 09 2014
Dickson on page 384 gives the Diophantine equation "24x^2 + 1 = y^2" and later states "y_{n+1} = 10y_n - y_{n-1}" where y_n is this sequence. - Michael Somos, Jun 19 2023

			Pell equation: n = 0: 1^2 - 24*0^2 = +1, n = 1: 5^2 - 6*(1*2)^2 = 1, n = 2: 49^2 - 6*(2*10)^2 = +1. - _Wolfdieter Lang_, Jun 26 2013
G.f. = 1 + 5*x + 49*x^2 + 485*x^3 + 4801*x^4 + 47525*x^5 + 470449*x^6 + ...

Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163-166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From N. J. A. Sloane, May 30 2012
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 384.
L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 374.
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).
V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.

T. D. Noe, Table of n, a(n) for n=0..200
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Leonhard Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.
Leonhard Euler, De solutione problematum diophanteorum per numeros integros, par. 18.
Tanya Khovanova, Recursive Sequences
Robert Phillips, Polynomials of the form 1+4ke+4ke^2, 2008.
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
Index entries for linear recurrences with constant coefficients, signature (10,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A004189, A001078, A046173, A046172, A036353, A138281, A004189.

I:=[1,5]; [n le 2 select I[n] else 10*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 10 2016

A001079 := proc(n)
    option remember;
    if n <= 1 then
        op(n+1,[1,5]) ;
    else
        10*procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A001079(n),n=0..20) ; # R. J. Mathar, Apr 30 2017

Table[(-1)^n Round[N[Cos[2 n ArcSin[Sqrt[3]]], 50]], {n, 0, 20}] (* Artur Jasinski, Oct 29 2008 *)
a[ n_] := ChebyshevT[n, 5]; (* Michael Somos, Aug 24 2014 *)
CoefficientList[Series[(1-5*x)/(1-10*x+x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 20 2017 *)
a[n_] := 3^n*Sum[(2/3)^k*Binomial[2*n, 2*k], {k,0,n}]; Flatten[Table[a[n], {n,0,18}]] (* Detlef Meya, May 21 2024 *)

{a(n) = subst(poltchebi(n), 'x, 5)}; /* Michael Somos, Sep 05 2006 */

{a(n) = real((5 + 2*quadgen(24))^n)}; /* Michael Somos, Sep 05 2006 */

{a(n) = n = abs(n); polsym(1 - 10*x + x^2, n)[n+1] / 2}; /* Michael Somos, Sep 05 2006 */

x='x+O('x^30); Vec((1-5*x)/(1-10*x+x^2)) \\ G. C. Greubel, Dec 20 2017

For all members x of the sequence, 6*x^2 -6 is a square. Limit_{n->infinity} a(n)/a(n-1) = 5 + 2*sqrt(6). - Gregory V. Richardson, Oct 13 2002
a(n) = T(n, 5) = (S(n, 10)-S(n-2, 10))/2 with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 10) = A004189(n+1).
a(n) = sqrt(1+24*A004189(n)^2) (cf. Richardson comment).
a(n)*a(n+3) - a(n+1)*a(n+2) = 240. - Ralf Stephan, Jun 06 2005
Chebyshev's polynomials T(n,x) evaluated at x=5.
G.f.: (1-5*x)/(1-10*x+x^2). - Simon Plouffe in his 1992 dissertation
a(n)= ((5+2*sqrt(6))^n + (5-2*sqrt(6))^n)/2.
a(-n) = a(n).
a(n+1) = 5*a(n) + 2*(6*a(n)^2-6)^(1/2) - Richard Choulet, Sep 19 2007
(sqrt(2)+sqrt(3))^(2*n)=a(n)+A001078(n)*sqrt(6). - Reinhard Zumkeller, Mar 12 2008
a(n+1) = 2*A054320(n) + 3*A138288(n). - Reinhard Zumkeller, Mar 12 2008
a(n) = cosh(2*n* arcsinh(sqrt(2))). - Herbert Kociemba, Apr 24 2008
a(n) = (-1)^n * cos(2*n* arcsin(sqrt(3))). - Artur Jasinski, Oct 29 2008
a(n) = cos(2*n* arccos(sqrt(3))). - Artur Jasinski, Sep 10 2016
a(n) = A142238(2n-1) = A041006(2n-1) = A041038(2n-1), for all n > 0. - M. F. Hasler, Feb 14 2009
2*a(n)^2 = 3*A122652(n)^2 + 2. - Charlie Marion, Feb 01 2013
E.g.f.: cosh(2*sqrt(6)*x)*exp(5*x). - Ilya Gutkovskiy, Sep 10 2016
From Peter Bala, Aug 17 2022: (Start)
a(n) = (1/2)^n * [x^n] ( 10*x + sqrt(1 + 96*x^2) )^n.
The g.f. A(x) satisfies A(2*x) = 1 + x*B'(x)/B(x), where B(x) = 1/sqrt(1 - 20*x + 4*x^2) is the g.f. of A098270.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p >= 3 and positive integers n and k.
Sum_{n >= 1} 1/(a(n) - 3/a(n)) = 1/4.
Sum_{n >= 1} (-1)^(n+1)/(a(n) + 2/a(n)) = 1/6.
Sum_{n >= 1} 1/(a(n)^2 - 3) = 1/4 - 1/sqrt(24). (End)
a(n) = 3^n*Sum_{k=0..n} (2/3)^k*binomial(2*n, 2*k). - Detlef Meya, May 21 2024

Chebyshev comments from Wolfdieter Lang, Nov 08 2002

A001570 Numbers k such that k^2 is centered hexagonal.

Original entry on oeis.org

1, 13, 181, 2521, 35113, 489061, 6811741, 94875313, 1321442641, 18405321661, 256353060613, 3570537526921, 49731172316281, 692665874901013, 9647591076297901, 134373609193269601, 1871582937629476513, 26067787517619401581, 363077442309042145621

Offset: 1

N. J. A. Sloane

Chebyshev T-sequence with Diophantine property. - Wolfdieter Lang, Nov 29 2002
a(n) = L(n,14), where L is defined as in A108299; see also A028230 for L(n,-14). - Reinhard Zumkeller, Jun 01 2005
Numbers x satisfying x^2 + y^3 = (y+1)^3. Corresponding y given by A001921(n)={A028230(n)-1}/2. - Lekraj Beedassy, Jul 21 2006
Mod[ a(n), 12 ] = 1. (a(n) - 1)/12 = A076139(n) = Triangular numbers that are one-third of another triangular number. (a(n) - 1)/4 = A076140(n) = Triangular numbers T(k) that are three times another triangular number. - Alexander Adamchuk, Apr 06 2007
Also numbers n such that RootMeanSquare(1,3,...,2*n-1) is an integer. - Ctibor O. Zizka, Sep 04 2008
a(n), with n>1, is the length of the cevian of equilateral triangle whose side length is the term b(n) of the sequence A028230. This cevian divides the side (2*x+1) of the triangle in two integer segments x and x+1. - Giacomo Fecondo, Oct 09 2010
For n>=2, a(n) equals the permanent of the (2n-2)X(2n-2) tridiagonal matrix with sqrt(12)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
Beal's conjecture would imply that set intersection of this sequence with the perfect powers (A001597) equals {1}. In other words, existence of a nontrivial perfect power in this sequence would disprove Beal's conjecture. - Max Alekseyev, Mar 15 2015
Numbers n such that there exists positive x with x^2 + x + 1 = 3n^2. - Jeffrey Shallit, Dec 11 2017
Given by the denominators of the continued fractions [1,(1,2)^i,3,(1,2)^{i-1},1]. - Jeffrey Shallit, Dec 11 2017
A near-isosceles integer-sided triangle with an angle of 2*Pi/3 is a triangle whose sides (a, a+1, c) satisfy Diophantine equation (a+1)^3 - a^3 = c^2. For n >= 2, the largest side c is given by a(n) while smallest and middle sides (a, a+1) = (A001921(n-1), A001922(n-1)) (see Julia link). - Bernard Schott, Nov 20 2022

			G.f. = x + 13*x^2 + 181*x^3 + 2521*x^4 + 35113*x^5 + 489061*x^6 + 6811741*x^7 + ...

E.-A. Majol, Note #2228, L'Intermédiaire des Mathématiciens, 9 (1902), pp. 183-185. - N. J. A. Sloane, Mar 03 2022
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..870 (terms 1..101 from T. D. Noe)
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
G. Julia, Triangles dont un angle mesure 120 degrés, Problème Capes, part C (in French).
Tanya Khovanova, Recursive Sequences
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
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.
Sociedad Magic Penny Patagonia, Leonardo en Patagonia
V. Thebault, Consecutive cubes with difference a square, Amer. Math. Monthly, 56 (1949), 174-175.
Eric Weisstein's World of Mathematics, Hex Number
Wikipedia, Beal's conjecture
Index entries for sequences related to Chebyshev polynomials.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (14,-1).

Bisection of A003500/4. Cf. A006051, A001921, A001922.
One half of odd part of bisection of A001075. First differences of A007655.
Cf. A077417 with companion A077416.
Row 14 of array A094954.
Cf. A076139, A076140, A102871.
A122571 is another version of the same sequence.
Row 2 of array A188646.
Cf. similar sequences listed in A238379.
Cf. A028231, which gives the corresponding values of x in 3n^2 = x^2 + x + 1.
Similar sequences of the type cosh((2*m+1)*arccosh(k))/k are listed in A302329. This is the case k=2.

[((2 + Sqrt(3))^(2*n - 1) + (2 - Sqrt(3))^(2*n - 1))/4: n in [1..50]]; // G. C. Greubel, Nov 04 2017

A001570:=-(-1+z)/(1-14*z+z**2); # Simon Plouffe in his 1992 dissertation.

NestList[3 + 7*#1 + 4*Sqrt[1 + 3*#1 + 3*#1^2] &, 0, 24] (* Zak Seidov, May 06 2007 *)
f[n_] := Simplify[(2 + Sqrt@3)^(2 n - 1) + (2 - Sqrt@3)^(2 n - 1)]/4; Array[f, 19] (* Robert G. Wilson v, Oct 28 2010 *)
a[c_, n_] := Module[{},
   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
   d := Denominator[Convergents[Sqrt[c], n p]];
   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
   Return[t];
  ] (* Complement of A041017 *)
a[12, 20] (* Gerry Martens, Jun 07 2015 *)
LinearRecurrence[{14, -1}, {1, 13}, 19] (* Jean-François Alcover, Sep 26 2017 *)
CoefficientList[Series[x (1-x)/(1-14x+x^2),{x,0,20}],x] (* Harvey P. Dale, Sep 18 2024 *)

{a(n) = real( (2 + quadgen( 12)) ^ (2*n - 1)) / 2}; /* Michael Somos, Feb 15 2011 */

a(n) = ((2 + sqrt(3))^(2*n - 1) + (2 - sqrt(3))^(2*n - 1)) / 4. - Michael Somos, Feb 15 2011
G.f.: x * (1 - x) / (1 -14*x + x^2). - Michael Somos, Feb 15 2011
Let q(n, x) = Sum_{i=0, n} x^(n-i)*binomial(2*n-i, i) then a(n) = q(n, 12). - Benoit Cloitre, Dec 10 2002
a(n) = S(n, 14) - S(n-1, 14) = T(2*n+1, 2)/2 with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x)=0, S(n, 14)=A007655(n+1) and T(n, 2)=A001075(n). - Wolfdieter Lang, Nov 29 2002
a(n) = A001075(n)*A001075(n+1) - 1 and thus (a(n)+1)^6 has divisors A001075(n)^6 and A001075(n+1)^6 congruent to -1 modulo a(n) (cf. A350916). - Max Alekseyev, Jan 23 2022
4*a(n)^2 - 3*b(n)^2 = 1 with b(n)=A028230(n+1), n>=0.
a(n)*a(n+3) = 168 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004
a(n) = 14*a(n-1) - a(n-2), a(0) = a(1) = 1. a(1 - n) = a(n) (compare A122571).
a(n) = 12*A076139(n) + 1 = 4*A076140(n) + 1. - Alexander Adamchuk, Apr 06 2007
a(n) = (1/12)*((7-4*sqrt(3))^n*(3-2*sqrt(3))+(3+2*sqrt(3))*(7+4*sqrt(3))^n -6). - Zak Seidov, May 06 2007
a(n) = A102871(n)^2+(A102871(n)-1)^2; sum of consecutive squares. E.g. a(4)=36^2+35^2. - Mason Withers (mwithers(AT)semprautilities.com), Jan 26 2008
a(n) = sqrt((3*A028230(n+1)^2 + 1)/4).
a(n) = A098301(n+1) - A001353(n)*A001835(n).
a(n) = A000217(A001571(n-1)) + A000217(A133161(n)), n>=1. - Ivan N. Ianakiev, Sep 24 2013
a(n)^2 = A001922(n-1)^3 - A001921(n-1)^3, for n >= 1. - Bernard Schott, Nov 20 2022
a(n) = 2^(2*n-3)*Product_{k=1..2*n-1} (2 - sin(2*Pi*k/(2*n-1))). Michael Somos, Dec 18 2022
a(n) = A003154(A101265(n)). - Andrea Pinos, Dec 19 2022

A033890 a(n) = Fibonacci(4*n + 2).

Original entry on oeis.org

1, 8, 55, 377, 2584, 17711, 121393, 832040, 5702887, 39088169, 267914296, 1836311903, 12586269025, 86267571272, 591286729879, 4052739537881, 27777890035288, 190392490709135, 1304969544928657, 8944394323791464, 61305790721611591, 420196140727489673

Offset: 0

N. J. A. Sloane

(x,y) = (a(n), a(n+1)) are solutions of (x+y)^2/(1+xy)=9, the other solutions are in A033888. - Floor van Lamoen, Dec 10 2001
This sequence consists of the odd-indexed terms of A001906 (whose terms are the values of x such that 5*x^2 + 4 is a square). The even-indexed terms of A001906 are in A033888. Limit_{n->infinity} a(n)/a(n-1) = phi^4 = (7 + 3*sqrt(5))/2. - Gregory V. Richardson, Oct 13 2002
General recurrence is a(n) = (a(1)-1)*a(n-1) - a(n-2), a(1) >= 4, lim_{n->infinity} a(n) = x*(k*x+1)^n, k = a(1) - 3, x = (1 + sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878. a(1)=5 gives A001834. a(1)=6 gives A030221. a(1)=7 gives A002315. a(1)=8 gives A033890. a(1)=9 gives A057080. a(1)=10 gives A057081. - Ctibor O. Zizka, Sep 02 2008
Indices of square numbers which are also 12-gonal. - Sture Sjöstedt, Jun 01 2009
For positive n, a(n) equals the permanent of the (2n) X (2n) tridiagonal matrix with 3's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
If we let b(0) = 0 and, for n >= 1, b(n) = A033890(n-1), then the sequence b(n) will be F(4n-2) and the first difference is L(4n) or A056854. F(4n-2) is also the ratio of golden spiral length (rounded to the nearest integer) after n rotations. L(4n) is also the pitch length ratio. See illustration in links. - Kival Ngaokrajang, Nov 03 2013
The aerated sequence (b(n))n>=1 = [1, 0, 8, 0, 55, 0, 377, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -5, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. - Peter Bala, Mar 22 2015
Solutions y of Pell equation x^2 - 5*y^2 = 4; corresponding x values are in A342710 (see A342709). - Bernard Schott, Mar 19 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
Nathan D. Cahill and Darren A. Narayan, Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants, Fib. Quart. 42, no. 3, Aug. 2004, pp. 216-221. See p. 219.
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
Tanya Khovanova, Recursive Sequences
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174.
Kival Ngaokrajang, Illustration of golden spiral length and pitch ratio
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (7,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000045, A001906, A100047.

Cf. A342709, A342710.

[Fibonacci(4*n +2): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

A033890 := proc(n)
    option remember;
    if n <= 1 then
        op(n+1,[1,8]);
    else
        7*procname(n-1)-procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Apr 30 2017

Table[Fibonacci[4n + 2], {n, 0, 14}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2008 *)
LinearRecurrence[{7, -1}, {1, 8}, 50] (* G. C. Greubel, Jul 13 2017 *)
a[n_] := (GoldenRatio^(2 (1 + 2 n)) - GoldenRatio^(-2 (1 + 2 n)))/Sqrt[5]
Table[a[n] // FullSimplify, {n, 0, 21}] (* Gerry Martens, Aug 20 2025 *)

PARI
```
a(n)=fibonacci(4*n+2);
```

G.f.: (1+x)/(1-7*x+x^2).
a(n) = 7*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=8.
a(n) = S(n,7) + S(n-1,7) = S(2*n,sqrt(9) = 3), where S(n,x) = U(n,x/2) are Chebyshev's polynomials of the 2nd kind. Cf. A049310. S(n,7) = A004187(n+1), S(n,3) = A001906(n+1).
a(n) = ((7+3*sqrt(5))^n - (7-3*sqrt(5))^n + 2*((7+3*sqrt(5))^(n-1) - ((7-3*sqrt(5))^(n-1)))) / (3*(2^n)*sqrt(5)). - Gregory V. Richardson, Oct 13 2002
Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then a(n) = (-1)^n*q(n, -9). - Benoit Cloitre, Nov 10 2002
a(n) = L(n,-7)*(-1)^n, where L is defined as in A108299; see also A049685 for L(n,+7). - Reinhard Zumkeller, Jun 01 2005
Define f(x,s) = s*x + sqrt((s^2-1)*x^2+1); f(0,s)=0. a(n) = f(a(n-1),7/2) + f(a(n-2),7/2). - Marcos Carreira, Dec 27 2006
a(n+1) = 8*a(n) - 8*a(n-1) + a(n-2); a(1)=1, a(2)=8, a(3)=55. - Sture Sjöstedt, May 27 2009
a(n) = A167816(4*n+2). - Reinhard Zumkeller, Nov 13 2009
a(n)=b such that (-1)^n*Integral_{0..Pi/2} (cos((2*n+1)*x))/(3/2-sin(x)) dx = c + b*log(3). - Francesco Daddi, Aug 01 2011
a(n) = A000045(A016825(n)). - Michel Marcus, Mar 22 2015
a(n) = A001906(2*n+1). - R. J. Mathar, Apr 30 2017
E.g.f.: exp(7*x/2)*(5*cosh(3*sqrt(5)*x/2) + 3*sqrt(5)*sinh(3*sqrt(5)*x/2))/5. - Stefano Spezia, Apr 14 2025
From Peter Bala, Jun 08 2025: (Start)
Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/9 [telescoping series: 3/(a(n) - 1/a(n)) = 1/Fibonacci(4*n+4) + 1/Fibonacci(4*n)].
Product_{n >= 1} (a(n) + 3)/(a(n) - 3) = 5/2 [telescoping product:
(a(n) + 3)/(a(n) - 3) = b(n)/b(n-1), where b(n) = (Lucas(4*n+4) - 3)/(Lucas(4*n+4) + 3)].
Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(9/5) [telescoping product:
(a(n) + 1)/(a(n) - 1) = c(n)/c(n-1) for n >= 1, where c(n) = Fibonacci(2*n+2)/Lucas(2*n+2)]. (End)
From Gerry Martens, Aug 20 2025: (Start)
a(n) = ((3 + sqrt(5))^(1 + 2*n) - (3 - sqrt(5))^(1 + 2*n)) / (2^(1 + 2*n)*sqrt(5)).
a(n) = Sum_{k=0..2*n} binomial(2*n + k + 1, 2*k + 1). (End)

A052918 a(0) = 1, a(1) = 5, a(n+1) = 5*a(n) + a(n-1).

Original entry on oeis.org

1, 5, 26, 135, 701, 3640, 18901, 98145, 509626, 2646275, 13741001, 71351280, 370497401, 1923838285, 9989688826, 51872282415, 269351100901, 1398627786920, 7262490035501, 37711077964425, 195817879857626

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A087130(n)^2 - 29*a(n-1)^2 = 4*(-1)^n, n >= 1. - Gary W. Adamson, Jul 01 2003, corrected Oct 07 2008, corrected by Jianing Song, Feb 01 2019
a(p-1) == 29^((p-1)/2) (mod p), for odd primes p. - Gary W. Adamson, Feb 22 2009 [See A087475 for more info about this congruence. - Jason Yuen, Apr 05 2025]
For more information about this type of recurrence, follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010
Binomial transform of A015523. - Johannes W. Meijer, Aug 01 2010
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 5's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
a(n) equals the number of words of length n on alphabet {0,1,...,5} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, Feb 15 2023: (Start)
Also called the 5-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 5 kinds of squares available. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 8.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 901
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
Milan Janjić, 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
Kai Wang, On k-Fibonacci Sequences And Infinite Series List of Results and Examples, 2020.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (5,1).

Row 5 of A073133, A172236, and A352361.

Cf. A087130, A099365 (squares), A100237, A175184 (Pisano periods), A201005 (prime subsequence).

a:=[1,5];; for n in [3..30] do a[n]:=5*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Oct 16 2019

I:=[1, 5]; [n le 2 select I[n] else 5*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 23 2013

R:=PowerSeriesRing(Integers(), 22); Coefficients(R!( 1/(1 - 5*x - x^2) )); // Marius A. Burtea, Oct 16 2019

spec := [S,{S=Sequence(Union(Z,Z,Z,Z,Z,Prod(Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..30);
a[0]:=1: a[1]:=5: for n from 2 to 26 do a[n]:=5*a[n-1]+a[n-2] od: seq(a[n], n=0..30); # Zerinvary Lajos, Jul 26 2006
with(combinat):a:=n->fibonacci(n,5):seq(a(n),n=1..30); # Zerinvary Lajos, Dec 07 2008

LinearRecurrence[{5, 1}, {1, 5}, 30] (* Vincenzo Librandi, Feb 23 2013 *)
Table[Fibonacci[n+1, 5], {n,0,30}] (* Vladimir Reshetnikov, May 08 2016 *)

Vec(1/(1-5*x-x^2)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2011

[lucas_number1(n,5,-1) for n in range(1, 22)] # Zerinvary Lajos, Apr 24 2009

G.f.: 1/(1 - 5*x - x^2).
a(3n) = A041047(5n), a(3n+1) = A041047(5n+3), a(3n+2) = 2*A041047(5n+4). - Henry Bottomley, May 10 2000
a(n) = Sum_{alpha=RootOf(-1+5*z+z^2)} (1/29)*(5+2*alpha)*alpha^(-1-n).
a(n-1) = (((5 + sqrt(29))/2)^n - ((5 - sqrt(29))/2)^n)/sqrt(29). - Gary W. Adamson, Jul 01 2003
a(n) = U(n, 5*i/2)*(-i)^n with i^2 = -1 and Chebyshev's U(n, x/2) = S(n, x) polynomials. See triangle A049310.
Let M = {{0, 1}, {1, 5}}, then a(n) is the lower-right term of M^n. - Roger L. Bagula, May 29 2005
a(n) = F(n, 5), the n-th Fibonacci polynomial evaluated at x = 5. - T. D. Noe, Jan 19 2006
a(n) = denominator of n-th convergent to [1, 4, 5, 5, 5, ...], for n > 0. Continued fraction [1, 4, 5, 5, 5, ...] = 0.807417596..., the inradius of a right triangle with legs 2 and 5. n-th convergent = A100237(n)/A052918(n), the first few being: 1/1, 4/5, 21/26, 109/135, 566/701, ... - Gary W. Adamson, Dec 21 2007
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 5*A097781(n), a(2n) = A097835(n).
Limit_{k->oo} a(n+k)/a(k) = (A087130(n) + a(n-1)*sqrt(29))/2.
Limit_{n->oo} A087130(n)/a(n-1) = sqrt(29). (End)
From L. Edson Jeffery, Jan 07 2012: (Start)
Define the 2 X 2 matrix A = {{1, 1}, {5, 4}}. Then:
a(n) is the upper-left term of (1/5)*(A^(n+2) - A^(n+1));
a(n) is the upper-right term of A^(n+1);
a(n) is the lower-left term of (1/5)*A^(n+1);
a(n) is the lower-right term of (Sum_{k=0..n} A^k). (End)
Sum_{n>=0} (-1)^n/(a(n)*a(n+1)) = (sqrt(29) - 5)/2. - Vladimir Shevelev, Feb 23 2013
G.f.: x/(1 - 5*x - x^2) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (m*k + 5 - m + x)/(1 + m*k*x) ) for arbitrary m (a telescoping series). - Peter Bala, May 08 2024

A098317 Decimal expansion of phi^3 = 2 + sqrt(5).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 02 2004

This sequence is also the decimal expansion of ((1+sqrt(5))/2)^3. - Mohammad K. Azarian, Apr 14 2008
This is the length/width ratio of a 4-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 10 2011
Its continued fraction is [4, 4, ...] (see A010709). - Robert G. Wilson v, Apr 10 2011

			4.23606797749978969640917366873127623544061835961152572427...

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
Alexey Stakhov, The mathematics of harmony: from Euclid to contemporary mathematics and computer science, World Scientific, Singapore, 2009, p. 657.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2

Cf. A001622, A014176, A098316, A098318.

Cf. A001076, A049310.

SetDefaultRealField(RealField(100)); 2+Sqrt(5); // G. C. Greubel, Jun 30 2019

RealDigits[N[2+Sqrt[5],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

sqrt(5)+2 \\ Charles R Greathouse IV, Mar 11 2012

numerical_approx(2+sqrt(5), digits=100) # G. C. Greubel, Jun 30 2019

2 plus the constant in A002163. - R. J. Mathar, Sep 02 2008
Equals 3 + 4*sin(Pi/10) = 1 + 4*cos(Pi/5) = 1 + 4*sin(3*Pi/10) = 3 + 4*cos(2*Pi/5) = 1 + csc(Pi/10). - Arkadiusz Wesolowski, Mar 11 2012
Equals lim_{n -> infinity} F(n+3)/F(n) = lim_{n -> infinity} (1 + 2*F(n+1)/F(n)) = 2 + sqrt(5), with F(n) = A000045(n). - Arkadiusz Wesolowski, Mar 11 2012
Equals exp(arcsinh(2)), since arcsinh(x) = log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013
Equals Sum_{n>=1} n/phi^n = phi/(phi-1)^2 = phi^3. - Richard R. Forberg, Jun 29 2014
Equals 1 + 2*phi, with phi = A001622, an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Dec 10 2022
c^n = A001076(n-1) + c * A001076(n); where c = 2 + sqrt(5). - Gary W. Adamson, Oct 09 2023
Equals lim_{n -> infinity} = S(n, 2*(-1 + 2*phi))/S(n-1, 2*(-1 + 2*phi)), with the S-Chebyshev polynomials (see A049310). See also the above limit formula with Fibonacci numbers. - Wolfdieter Lang, Nov 15 2023

Title expanded to include observation from Mohammad K. Azarian by Charles R Greathouse IV, Mar 11 2012

A007655 Standard deviation of A007654.

Original entry on oeis.org

0, 1, 14, 195, 2716, 37829, 526890, 7338631, 102213944, 1423656585, 19828978246, 276182038859, 3846719565780, 53577891882061, 746243766783074, 10393834843080975, 144767444036350576, 2016350381665827089, 28084137899285228670, 391161580208327374291, 5448177985017298011404

Offset: 1

N. J. A. Sloane

a(n) corresponds also to one-sixth the area of Fleenor-Heronian triangle with middle side A003500(n). - Lekraj Beedassy, Jul 15 2002
a(n) give all (nontrivial, integer) solutions of Pell equation b(n+1)^2 - 48*a(n+1)^2 = +1 with b(n+1)=A011943(n), n>=0.
For n>=3, a(n) equals the permanent of the (n-2) X (n-2) tridiagonal matrix with 14's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,13}. - Milan Janjic, Jan 25 2015
6*a(n)^2 = 6*S(n-1, 14)^2 is the triangular number Tri((T(n, 7) - 1)/2) with Tri = A000217 and T = A053120. This is instance k = 3 of the general k-identity given in a comment to A001109. - Wolfdieter Lang, Feb 01 2016

			G.f. = x^2 + 14*x^3 + 195*x^4 + 2716*x^5 + 37829*x^6 + 526890*x^7 + ...

D. A. Benaron, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 1..874 (terms 1..100 from T. D. Noe)
R. Flórez, R. A. Higuita and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
D. S. Hale, 3165. Perfect Squares of the Form 48n^2+1, Math. Gaz., Oct. 1966, page 307.
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
Murray S. Klamkin, Perfect Squares of the Form (m^2 - 1)a_n^2 + t, Math. Mag., 1969, page 111.
E. K. Lloyd, The standard deviation of 1, 2, ..., n, Pell's equation and rational triangles, The Mathematical Gazette, Vol. 81, No. 491 (Jul., 1997), pp. 231-243.
Dino Lorenzini and Z. Xiang, Integral points on variable separated curves, Preprint 2016.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (14,-1).

Cf. A000217, A001570, A003500, A011922, A011943, A011945, A028230, A046184, A049310, A053120, A055793, A067900, A098301, A101950, A103974.
Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), this sequence (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).
Cf. A323182.

m:=7;; a:=[0,1];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 23 2019

[n le 2 select n-1 else 14*Self(n-1)-Self(n-2): n in [1..70]]; // Vincenzo Librandi, Feb 02 2016

0,seq(orthopoly[U](n,7),n=0..30); # Robert Israel, Feb 04 2016

Table[GegenbauerC[n, 1, 7], {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
LinearRecurrence[{14,-1}, {0,1}, 20] (* Vincenzo Librandi, Feb 02 2016 *)
ChebyshevU[Range[21] -2, 7] (* G. C. Greubel, Dec 23 2019 *)
Table[Sum[Binomial[n, 2 k - 1]*7^(n - 2 k + 1)*48^(k - 1), {k, 1, n}], {n, 0, 15}] (* Horst H. Manninger, Jan 16 2022 *)

concat(0, Vec((x^2/(1-14*x+x^2) + O(x^30)))) \\ Michel Marcus, Feb 02 2016

vector(21, n, polchebyshev(n-2, 2, 7) ) \\ G. C. Greubel, Dec 23 2019

[lucas_number1(n,14,1) for n in range(0,20)] # Zerinvary Lajos, Jun 25 2008

[chebyshev_U(n,7) for n in (-1..20)] # G. C. Greubel, Dec 23 2019

a(n) = 14*a(n-1) - a(n-2).
G.f.: x^2/(1-14*x+x^2).
a(n+1) ~ 1/24*sqrt(3)*(2 + sqrt(3))^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 2002
a(n+1) = S(n-1, 14), n>=0, with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. S(-1, x) := 0. See A049310.
a(n+1) = ( (7+4*sqrt(3))^n - (7-4*sqrt(3))^n )/(8*sqrt(3)).
a(n+1) = sqrt((A011943(n)^2 - 1)/48), n>=0.
Chebyshev's polynomials U(n-2, x) evaluated at x=7.
a(n) = A001353(2n)/4. - Lekraj Beedassy, Jul 15 2002
4*a(n+1) + A046184(n) = A055793(n+2) + A098301(n+1) 4*a(n+1) + A098301(n+1) + A055793(n+2) = A046184(n+1) (4*a(n+1))^2 = A098301(2n+1) (conjectures). - Creighton Dement, Nov 02 2004
(4*a(n))^2 = A103974(n)^2 - A011922(n-1)^2. - Paul D. Hanna, Mar 06 2005
From Mohamed Bouhamida, May 26 2007: (Start)
a(n) = 13*( a(n-1) + a(n-2) ) - a(n-3).
a(n) = 15*( a(n-1) - a(n-2) ) + a(n-3). (End)
a(n) = b such that (-1)^n/4*Integral_{x=-Pi/2..Pi/2} (sin((2*n-2)*x))/(2-sin(x)) dx = c+b*log(3). - Francesco Daddi, Aug 02 2011
a(n+2) = Sum_{k=0..n} A101950(n,k)*13^k. - Philippe Deléham, Feb 10 2012
Product {n >= 1} (1 + 1/a(n)) = 1/3*(3 + 2*sqrt(3)). - Peter Bala, Dec 23 2012
Product {n >= 2} (1 - 1/a(n)) = 1/7*(3 + 2*sqrt(3)). - Peter Bala, Dec 23 2012
a(n) = (A028230(n) - A001570(n))/2. - Richard R. Forberg, Nov 14 2013
E.g.f.: 1 - exp(7*x)*(12*cosh(4*sqrt(3)*x) - 7*sqrt(3)*sinh(4*sqrt(3)*x))/12. - Stefano Spezia, Dec 11 2022

Chebyshev comments from Wolfdieter Lang, Nov 08 2002

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