A079962 -id:A079962 - cp's OEIS frontend

A001077 Numerators of continued fraction convergents to sqrt(5).

1, 2, 9, 38, 161, 682, 2889, 12238, 51841, 219602, 930249, 3940598, 16692641, 70711162, 299537289, 1268860318, 5374978561, 22768774562, 96450076809, 408569081798, 1730726404001, 7331474697802, 31056625195209

Offset: 0

N. J. A. Sloane

a(2*n+1) with b(2*n+1) := A001076(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation a^2 - 5*b^2 = -1.
a(2*n) with b(2*n) := A001076(2*n), n >= 1, give all (positive integer) solutions to Pell equation a^2 - 5*b^2 = +1 (see Emerson reference).
Bisection: a(2*n) = T(n,9) = A023039(n), n >= 0 and a(2*n+1) = 2*S(2*n, 2*sqrt(5)) = A075796(n+1), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. See A053120, resp. A049310.
From Greg Dresden, May 21 2023: (Start)
For n >= 2, 8*a(n) is the number of ways to tile this T-shaped figure of length n-1 with four colors of squares and one color of domino; shown here is the figure of length 5 (corresponding to n=6), and it has 8*a(6) = 23112 different tilings.
    _
   || _  
   |||_|||
   |_|
(End)

			1  2  9  38  161  (A001077)
-, -, -, --, ---, ...
0  1  4  17   72  (A001076)
1 + 2*x + 9*x^2 + 38*x^3 + 161*x^4 + 682*x^5 + 2889*x^6 + 12238*x^7 + ... - _Michael Somos_, Aug 11 2009

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. 282.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242, Ex. 1, pp. 237-238.
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000032, A001076, A023039, A049629, A052924, A078343, A164581, A179237, A180148, A329723, A374439.

I:=[1, 2]; [n le 2 select I[n] else 4*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 19 2017

A001077:=(-1+2*z)/(-1+4*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation
with(combinat): a:=n->fibonacci(n+1, 4)-2*fibonacci(n, 4): seq(a(n), n=0..30); # Zerinvary Lajos, Apr 04 2008

LinearRecurrence[{4, 1}, {1, 2}, 30]
Join[{1},Numerator[Convergents[Sqrt[5],30]]] (* Harvey P. Dale, Mar 23 2016 *)
CoefficientList[Series[(1-2*x)/(1-4*x-x^2), {x, 0, 30}], x] (* G. C. Greubel, Dec 19 2017 *)
LucasL[3*Range[0,30]]/2 (* Rigoberto Florez, Apr 03 2019 *)
a[ n_] := LucasL[n, 4]/2; (* Michael Somos, Nov 02 2021 *)

{a(n) = fibonacci(3*n) / 2 + fibonacci(3*n - 1)}; /* Michael Somos, Aug 11 2009 */

a(n)=if(n<2,n+1,my(t=4);for(i=1,n-2,t=4+1/t);numerator(2+1/t)) \\ Charles R Greathouse IV, Dec 05 2011

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

[lucas_number2(n,4,-1)/2 for n in range(0, 30)] # Zerinvary Lajos, May 14 2009

G.f.: (1-2*x)/(1-4*x-x^2).
a(n) = 4*a(n-1) + a(n-2), a(0)=1, a(1)=2.
a(n) = ((2 + sqrt(5))^n + (2 - sqrt(5))^n)/2.
a(n) = A014448(n)/2.
Limit_{n->infinity} a(n)/a(n-1) = phi^3 = 2 + sqrt(5). - Gregory V. Richardson, Oct 13 2002
a(n) = ((-i)^n)*T(n, 2*i), with T(n, x) Chebyshev's polynomials of the first kind A053120 and i^2 = -1.
Binomial transform of A084057. - Paul Barry, May 10 2003
E.g.f.: exp(2x)cosh(sqrt(5)x). - Paul Barry, May 10 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*5^k*2^(n-2k). - Paul Barry, Nov 15 2003
a(n) = 4*a(n-1) + a(n-2) when n > 2; a(1) = 1, a(2) = 2. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 25 2004
a(n) = A001076(n+1) - 2*A001076(n) = A097924(n) - A015448(n+1); a(n+1) = A097924(n) + 2*A001076(n) = A097924(n) + 2(A048876(n) - A048875(n)). - Creighton Dement, Mar 19 2005
a(n) = F(3*n)/2 + F(3*n-1) where F() = Fibonacci numbers A000045. - Gerald McGarvey, Apr 28 2007
a(n) = A000032(3*n)/2.
For n >= 1: a(n) = (1/2)*Fibonacci(6*n)/Fibonacci(3*n) and a(n) = integer part of (2 + sqrt(5))^n. - Artur Jasinski, Nov 28 2011
a(n) = Sum_{k=0..n} A201730(n,k)*4^k. - Philippe Deléham, Dec 06 2011
a(n) = A001076(n) + A015448(n). - R. J. Mathar, Jul 06 2012
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(5*k-4)/(x*(5*k+1) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
a(n) is the (1,1)-entry of the matrix W^n with W=[2, sqrt(5); sqrt(5), 2]. - Carmine Suriano, Mar 21 2014
From Rigoberto Florez, Apr 03 2019: (Start)
a(n) = A099919(n) + A049651(n) if n > 0.
a(n) = 1 + Sum_{k=0..n-1} L(3*k + 1) if n >= 0, L(n) = n-th Lucas number (A000032). (End)
From Christopher Hohl, Aug 22 2021: (Start)
For n >= 2, a(2n-1) = A079962(6n-9) + A079962(6n-3).
For n >= 1, a(2n) = sqrt(20*A079962(6n-3)^2 + 1). (End)
a(n) = Sum_{k=0..n-2} A168561(n-2,k)*4^k + 2 * Sum_{k=0..n-1} A168561(n-1,k)*4^k, n>0. - R. J. Mathar, Feb 14 2024
a(n) = 4^n*Sum_{k=0..n} A374439(n, k)*(-1/4)^k. - Peter Luschny, Jul 26 2024
From Peter Bala, Jul 08 2025: (Start)
The following series telescope:
Sum_{n >= 1} 1/(a(n) + 5*(-1)^(n+1)/a(n)) = 3/8, since 1/(a(n) + 5*(-1)^(n+1)/a(n)) = b(n) - b(n+1), where b(n) = (1/4) * (a(n) + a(n-1)) / (a(n)*a(n-1)).
Sum_{n >= 1} (-1)^(n+1)/(a(n) + 5*(-1)^(n+1)/a(n)) = 1/8, since 1/(a(n) + 5*(-1)^(n+1)/a(n)) = c(n) + c(n+1), where c(n) = (1/4) * (a(n) - a(n-1)) / (a(n)*a(n-1)). (End)

Chebyshev comments from Wolfdieter Lang, Jan 10 2003

A115730 a(n) = a(n-3) + A001654(n-1) with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 2, 6, 16, 42, 110, 289, 756, 1980, 5184, 13572, 35532, 93025, 243542, 637602, 1669264, 4370190, 11441306, 29953729, 78419880, 205305912, 537497856, 1407187656, 3684065112, 9645007681, 25250957930, 66107866110

Offset: 0

Roger L. Bagula, Mar 13 2006

The a(n+1) represent the Ca2 and Ze4 sums of the Golden Triangle A180662. Furthermore the a(3*n) represent the Ze1 (terms doubled) and Ca3 sums of the Golden triangle. See A180662 for more information about these and other triangle sums.

			G.f. = x^2 + 2*x^3 + 6*x^4 + 16*x^5 + 42*x^6 + 110*x^7 + 289*x^8 + ... - _Michael Somos_, Sep 05 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,0,-2,-2,1).

Cf. A000045, A001654, A064831, A079962, A180662, A180664, A180665, A180666, A182890.

function A115730(n)
  if n lt 3 then return Floor(n/2);
  else return A115730(n-3) + Fibonacci(n-1)*Fibonacci(n);
  end if; return A115730;
end function;
[A115730(n): n in [0..40]]; // G. C. Greubel, Jan 20 2022

nmax:=31: with(combinat): for n from 0 to nmax do A001654(n):=fibonacci(n) * fibonacci(n+1) od: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):=a(n-3) + A001654(n-1) od: seq(a(n),n=0..nmax);

LinearRecurrence[{2,2,0,-2,-2,1}, {0,0,1,2,6,16}, 40] (* modified by G. C. Greubel, Jan 20 2022 *)
a[ n_] := Floor[(2*Fibonacci[2*n+1] + Fibonacci[2*n+2] + 2)/20]; (* Michael Somos, Sep 05 2023 *)

{a(n) = (2*fibonacci(2*n+1) + fibonacci(2*n+2) + 2)\20}; /* Michael Somos, Sep 05 2023 */

U=chebyshev_U
def A115730(n): return (1/60)*((-1)^n*(6 - 5*U(n, 1/2) + 10*U(n-1, 1/2)) - (10 - 9*U(n, 3/2) + 6*U(n-1, 3/2)))
[A115730(n) for n in (0..40)] # G. C. Greubel, Jan 20 2022

a(n) = -floor(g(Fibonacci(n+1))) where g(x) = (1-x^2)^2/(-4*x^2).
G.f.: x^2/( (1-x)*(1+x)*(1+x+x^2)*(1-3*x+x^2) ). - R. J. Mathar, Jun 20 2015
a(n) - a(n-2) = A182890(n-1). - R. J. Mathar, Jun 20 2015
a(n) = (1/60)*((-1)^n*(6 - 5*ChebyshevU(n, 1/2) + 10*ChebyshevU(n-1, 1/2)) - (10 - 9*ChebyshevU(n, 3/2) + 6*ChebyshevU(n-1, 3/2))). - G. C. Greubel, Jan 20 2022
a(n) = floor((2*Fibonacci(2*n+1) + Fibonacci(2*n+2) + 2)/20). - Michael Somos, Sep 05 2023

Corrected and information added by Johannes W. Meijer, Sep 22 2010

Edited by Editors-in-Chief. - N. J. A. Sloane, Jun 20 2015

A004697 a(n) = floor(Fibonacci(n)/4).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 5, 8, 13, 22, 36, 58, 94, 152, 246, 399, 646, 1045, 1691, 2736, 4427, 7164, 11592, 18756, 30348, 49104, 79452, 128557, 208010, 336567, 544577, 881144, 1425721, 2306866, 3732588, 6039454

Offset: 0

N. J. A. Sloane

Partial sums of A079962. - Mircea Merca, Jan 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1,1,0,-1).

See A000045 for the Fibonacci numbers.

Cf. A079962.

[Floor(Fibonacci(n)/4): n in [0..40]]; // Vincenzo Librandi, Jul 09 2012

A004697 := proc(n) floor(combinat[fibonacci](n)/4) ; end proc:

CoefficientList[Series[x^5/((1-x)*(1-x-x^2)*(1+x^2+x^4)),{x,0,50}],x] (* Vincenzo Librandi, Jul 09 2012 *)
Floor[Fibonacci[Range[0,50]]/4] (* or *) LinearRecurrence[ {2,-1,1,-1,1,0,-1},{0,0,0,0,0,1,2},50] (* Harvey P. Dale, Dec 05 2012 *)

vector(50, n, n--; fibonacci(n)\4) \\ G. C. Greubel, Oct 09 2018

G.f.: x^5 / ((1-x)*(1-x-x^2)*(1+x^2+x^4)).
From Mircea Merca, Jan 04 2011: (Start)
a(n) = floor(Fibonacci(n)/4).
a(n) = ceiling(Fibonacci(n)/4-3/4).
a(n) = round(Fibonacci(n)/4-3/8).
a(n) = Sum_{k=1..n-2} round(Fibonacci(n)/4).
a(n) = a(n-6) + Fibonacci(n-3), n > 5. (End)
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-7). - R. J. Mathar, Jan 08 2011
a(n) = (A000045(n) - A079343(n))/4. - R. J. Mathar, Jul 14 2012

A207832 Numbers x such that 20*x^2 + 1 is a perfect square.

Original entry on oeis.org

0, 2, 36, 646, 11592, 208010, 3732588, 66978574, 1201881744, 21566892818, 387002188980, 6944472508822, 124613502969816, 2236098580947866, 40125160954091772, 720016798592704030

Offset: 0

Gary Detlefs, Feb 20 2012

Denote as {a,b,c,d} the second-order linear recurrence a(n) = c*a(n-1) + d*a(n-2) with initial terms a, b. The following sequences and recurrence formulas are related to integer solutions of k*x^2 + 1 = y^2.
.
   k             x                        y
   -  -----------------------  -----------------------
A001542 {0,2,6,-1}       A001541 {1,3,6,-1}
A001353 {0,1,4,-1}       A001075 {1,2,4,-1}
A060645 {0,4,18,-1}      A023039 {1,9,18,-1}
A001078 {0,2,10,-1}      A001079 {1,5,10,-1}
A001080 {0,3,16,-1}      A001081 {1,8,16,-1}
A001109 {0,1,6,-1}       A001541 {1,3,6,-1}
A084070 {0,1,38,-1}      A078986 {1,19,38,-1}
A001084 {0,3,20,-1}      A001085 {1,10,20,-1}
A011944 {0,2,14,-1}      A011943 {1,7,14,-1}
A075871 {0,180,1298,-1}  A114047 {1,649,1298,-1}
A068204 {0,4,30,-1}      A069203 {1,15,30,-1}
A001090 {0,1,8,-1}       A001091 {1,4,8,-1}
A121740 {0,8,66,-1}      A099370 {1,33,66,-1}
A202299 {0,4,34,-1}      A056771 {1,17,34,-1}
A174765 {0,39,340,-1}    A114048 {1,179,340,-1}
a(n)    {0,2,18,-1}      A023039 {1,9,18,-1}
A174745 {0,12,110,-1}    A114049 {1,55,110,-1}
A174766 {0,42,394,-1}    A114050 {1,197,394,-1}
A174767 {0,5,48,-1}      A114051 {1,24,48,-1}
A004189 {0,1,10,-1}      A001079 {1,5,10,-1}
A174768 {0,10,102,-1}    A099397 {1,51,102,-1}
The sequence of the c parameter is listed in A180495.

Bruno Berselli, Table of n, a(n) for n = 0..500
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Index entries for linear recurrences with constant coefficients, signature (18,-1).

Cf. A023039, A049660, A079962.

m:=16; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x/(1-18*x+x^2))); // Bruno Berselli, Jun 19 2019

readlib(issqr):for x from 1 to 720016798592704030 do if issqr(20*x^2+1) then print(x) fi od;

LinearRecurrence[{18, -1}, {0, 2}, 16] (* Bruno Berselli, Feb 21 2012 *)
Table[2 ChebyshevU[-1 + n, 9], {n, 0, 16}]  (* Herbert Kociemba, Jun 05 2022 *)

makelist(expand(((2+sqrt(5))^(2*n)-(2-sqrt(5))^(2*n))/(4*sqrt(5))), n, 0, 15); /* Bruno Berselli, Jun 19 2019 */

a(n) = 18*a(n-1) - a(n-2).
From Bruno Berselli, Feb 21 2012: (Start)
G.f.: 2*x/(1-18*x+x^2).
a(n) = -a(-n) = 2*A049660(n) = ((2 + sqrt(5))^(2*n)-(2 - sqrt(5))^(2*n))/(4*sqrt(5)). (End)
a(n) = Fibonacci(6*n)/4. - Bruno Berselli, Jun 19 2019
For n>=1, a(n) = A079962(6n-3). - Christopher Hohl, Aug 22 2021

A118923 Triangle T(n,k) built by placing T(n,0)=A000012(n) in the left edge, T(n,n)=A079978(n) on the right edge and filling the body with the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 3, 3, 2, 0, 1, 4, 6, 5, 2, 0, 1, 5, 10, 11, 7, 2, 1, 1, 6, 15, 21, 18, 9, 3, 0, 1, 7, 21, 36, 39, 27, 12, 3, 0, 1, 8, 28, 57, 75, 66, 39, 15, 3, 1, 1, 9, 36, 85, 132, 141, 105, 54, 18, 4, 0, 1, 10, 45, 121, 217, 273, 246, 159, 72, 22, 4, 0, 1, 11, 55, 166

Offset: 0

Alford Arnold, May 05 2006

The fourth diagonal is 1, 2, 5, 11, 21, ..., which is 1 + A000292. The fifth diagonal is 0, 2, 7, 18, 39, 75, 132, 217, 338, 504, 725, 1012, ..., which is A051743.
The array A007318 is generated by placing A000012 on both edges with the same Pascal-like recurrence, and the array A059259 uses edges defined by A000012 and A059841. - R. J. Mathar, Jan 21 2008
From Michael A. Allen, Nov 30 2021: (Start)
T(n,n-k) is the (n,k)-th entry of the (1/(1-x^3), x/(1-x)) Riordan array.
Sums of rows give A077947.
Sums of antidiagonals give A079962. (End)

			The table begins
  1
  1  0
  1  1  0
  1  2  1  1
  1  3  3  2  0
  1  4  6  5  2  0
  1  5 10 11  7  2  1
  1  6 15 21 18  9  3  0

Cf. A000292, A079978, A008620, A079998, A051743, A077947.

Generalization of A007318, A059259.

A000012 := proc(n) 1 ; end: A079978 := proc(n) if n mod 3 = 0 then 1; else 0 ; fi ; end: A118923 := proc(n,k) if k = 0 then A000012(n); elif k = n then A079978(n) ; else A118923(n-1,k)+A118923(n-1,k-1) ; fi ; end: for n from 0 to 15 do for k from 0 to n do printf("%d, ",A118923(n,k)) ; od: od: # R. J. Mathar, Jan 21 2008

Flatten@Table[CoefficientList[Series[1/((1 + x*y + x^2*y^2)(1 - x - x*y)), {x, 0, 23}, {y, 0, 11}], {x, y}][[n + 1, k + 1]], {n, 0, 11}, {k, 0, n}] (* Michael A. Allen, Nov 30 2021 *)

From Michael A. Allen, Nov 30 2021: (Start)
For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/3)} binomial(n-3*j,n-k)/(n-3*j).
G.f.: 1/((1+x*y+(x*y)^2)*(1-x-x*y)). (End)

Edited and extended by R. J. Mathar, Jan 21 2008

Offset changed by Michael A. Allen, Nov 30 2021

A349840 The number of compositions of n using elements from the set {1,3,5,7,8}.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 22, 35, 56, 91, 147, 238, 385, 623, 1009, 1632, 2640, 4272, 6912, 11184, 18096, 29280, 47377, 76657, 124033, 200690, 324723, 525413, 850136, 1375549, 2225686, 3601235, 5826920, 9428155

Offset: 0

Michael A. Allen, Dec 05 2021

Number of ways to tile an n-board (an n X 1 array of 1 X 1 cells) using squares, trominoes, pentominoes, heptominoes, and octominoes.
Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-1,0,2,4,6,7} for all i=1,...,n.
a(n) gives the sums of the antidiagonals of A349839.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
K. Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,0,1,1).

Sums of antidiagonals of triangles in the same family as A349839: A000045, A006498, A079962, A349843.

CoefficientList[Series[1/(1-x-x^3-x^5-x^7-x^8),{x,0,35}],x]

a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-7) + a(n-8) + delta(n,0), a(n<0)=0 (where delta(i,j) is the Kronecker delta).
a(n) = a(n-1) + a(n-2) + a(n-8) - a(n-9) - a(n-10) + delta(n,0) - delta(n,2), a(n<0)=0.
G.f.: 1/(1-x-x^3-x^5-x^7-x^8).

A349843 Expansion of (1 - x^2)/((1 - x^10)*(1 - x - x^2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 56, 90, 145, 235, 380, 615, 995, 1610, 2605, 4215, 6821, 11036, 17856, 28892, 46748, 75640, 122388, 198028, 320416, 518444, 838861, 1357305, 2196165, 3553470, 5749635, 9303105

Offset: 0

Michael A. Allen, Dec 13 2021

The number of compositions of n using elements from the set {1,3,5,7,9,10}.
Number of ways to tile an n-board (an n X 1 array of 1 X 1 cells) using squares, trominoes, pentominoes, heptominoes, nonominoes, and decominoes.
Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-1,0,2,4,6,8,9} for all i=1,...,n.
a(n) gives the sums of the antidiagonals of A349841.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
K. Edwards and M. A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,0,1,0,1,1).

Sums of antidiagonals of triangles in the same family as A349841: A000045, A006498, A079962, A349840.

CoefficientList[Series[1/(1-x-x^3-x^5-x^7-x^9-x^10), {x, 0, 35}], x]

a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-7) + a(n-9) + a(n-10) + delta(n,0), a(n<0)=0.
a(n) = a(n-1) + a(n-2) + a(n-10) - a(n-11) - a(n-12) + delta(n,0) - delta(n,2), a(n<0)=0.
G.f.: 1/(1-x-x^3-x^5-x^7-x^9-x^10).

A115605 Expansion of -x^2(2 + x - 2x^2 - x^3 + 2x^4) / ( (x-1)(1+x)(1 + x + x^2)(x^2 - x + 1)(x^2 + 4x - 1)*(x^2 - x - 1) ).

Original entry on oeis.org

0, 0, 2, 7, 31, 128, 549, 2315, 9826, 41594, 176242, 746496, 3162334, 13395658, 56745250, 240376201, 1018250793, 4313378176, 18271765435, 77400436781, 327873517634, 1388894499108, 5883451527348, 24922700587008

Offset: 0

Roger L. Bagula, Mar 13 2006

Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1,0,1,-3,-6,3,1).

Cf. A000045, A079962.

A000035 := proc(n)
        n mod 2 ;
end proc:
A061347 := proc(n)
        op((n mod 3)+1,[-2,1,1]) ;
end proc:
A001076 := proc(n)
        option remember;
        if n <=1 then
                n;
        else
                4*procname(n-1)+procname(n-2) ;
        end if;
end proc:
A039834 := proc(n)
        (-1)^(n+1)*combinat[fibonacci](n) ;
end proc:
A087204 := proc(n)
        op((n mod 6)+1,[2,1,-1,-2,-1,1]) ;
end proc:
A115605 := proc(n)
        -A000035(n+1)/6 +A061347(n+2)/12 + A001076(n+1)/10 +3*A039834(n+1)/20 -A087204(n)/12 ;
end proc: # R. J. Mathar, Dec 16 2011

LinearRecurrence[{3,6,-3,-1,0,1,-3,-6,3,1},{0,0,2,7,31,128,549,2315,9826,41594},30] (* Harvey P. Dale, Dec 16 2011 *)

concat([0,0],Vec((2+x-2*x^2-x^3+2*x^4)/((1-x)*(1+x)*(1+x+x^2)*(x^2-x+1)*(x^2+4*x-1)*(x^2-x-1))+O(x^99))) \\ Charles R Greathouse IV, Sep 27 2012

Lim_{n->infinity} a(n+1)/a(n) = phi^3 = A098317.

a(n) = -A000035(n+1)/6 +A061347(n+2)/12 +A001076(n+1)/10 +3*A039834(n+1)/20 -A087204(n)/12. - R. J. Mathar, Dec 16 2011

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A001077 Numerators of continued fraction convergents to sqrt(5).

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A115730 a(n) = a(n-3) + A001654(n-1) with a(0)=0, a(1)=0 and a(2)=1.

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A004697 a(n) = floor(Fibonacci(n)/4).

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A207832 Numbers x such that 20*x^2 + 1 is a perfect square.

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A118923 Triangle T(n,k) built by placing T(n,0)=A000012(n) in the left edge, T(n,n)=A079978(n) on the right edge and filling the body with the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

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A349840 The number of compositions of n using elements from the set {1,3,5,7,8}.

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A115605 Expansion of -x^2(2 + x - 2x^2 - x^3 + 2x^4) / ( (x-1)(1+x)(1 + x + x^2)(x^2 - x + 1)(x^2 + 4x - 1)*(x^2 - x - 1) ).

A227047 Expansion of x^2(1+x^2) / ( (x^2-x+1)(-x^2-x+1)*(1+x+x^2) ).