A125816 a(n) = ((1+sqrt(13))^n + (1-sqrt(13))^n)/2.

1, 1, 14, 40, 248, 976, 4928, 21568, 102272, 463360, 2153984, 9868288, 45584384, 209588224, 966189056, 4447436800, 20489142272, 94347526144, 434564759552, 2001299832832, 9217376780288, 42450351554560, 195509224472576

Offset

1,3

Comments

Binomial transform of A001022(powers of 13), with interpolated zeros. - Philippe Deléham, Dec 20 2007

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,12).

Formula

From Philippe Deléham, Dec 12 2006: (Start)

a(n) = 2*a(n-1) + 12*a(n-2), with a(0)=a(1)=1.

G.f.: (1-x)/(1-2*x-12*x^2). (End)

a(n) = Sum_{k=0..n} A098158(n,k)*13^(n-k). - Philippe Deléham, Dec 20 2007

If p[1]=1, and p[i]=13, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1,(i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n+1)=det A. - Milan Janjic, Apr 29 2010

Mathematica

Expand[Table[((1+Sqrt[13])^n +(1-Sqrt[13])^n)/(2), {n, 0, 30}]] (* Artur Jasinski *)
LinearRecurrence[{2, 12}, {1, 1}, 30] (* G. C. Greubel, Aug 02 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-x)/(1-2*x-12*x^2)) \\ G. C. Greubel, Aug 02 2019
(Magma) I:=[1, 1]; [n le 2 select I[n] else 2*Self(n-1) +12*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019
(Sage) ((1-x)/(1-2*x-12*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019
(GAP) a:=[1, 1];; for n in [3..30] do a[n]:=2*a[n-1]+12*a[n-2]; od; a; # G. C. Greubel, Aug 02 2019

Crossrefs

Cf. A127262. First differences of A091914.

Sequence in context: A069126 A124707 A126368 * A105869 A216298 A056034

Adjacent sequences: A125813 A125814 A125815 * A125817 A125818 A125819

Keyword

nonn

Author

Artur Jasinski, Dec 10 2006

Status

approved