A125818 a(n) = ((1 + 3*sqrt(2))^n + (1 - 3*sqrt(2))^n)/2.

1, 1, 19, 55, 433, 1801, 10963, 52543, 291457, 1476145, 7907059, 40908583, 216237169, 1127920249, 5931872371, 31038388975, 162918608257, 853489829089, 4476595998547, 23462519091607, 123027170158513, 644917164874345, 3381296222443411, 17726184247750687

Offset

1,3

Comments

Binomial transform of [1, 0, 18, 0, 324, 0, 5832, 0, 104976, 0, ...] =: powers of 18 (A001027) with interpolated zeros. - Philippe Deléham, Dec 02 2008

a(n-1) is the number of compositions of n when there are 1 type of 1 and 18 types of other natural numbers. - Milan Janjic, Aug 13 2010

Links

T. D. Noe, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (2, 17).

Formula

From Philippe Deléham, Dec 12 2006: (Start)

a(n) = 2*a(n-1) + 17*a(n-2), with a(0)=a(1)=1.

G.f.: (1-x)/(1-2*x-17*x^2). (End)

a(n) = Sum_{k=0..n} A098158(n,k)*18^(n-k). - Philippe Deléham, Dec 26 2007

If p[1]=1, and p[i]=18, (i>1), and if A is Hessenberg matrix of order n If p[1]=1, and p[i]=18, (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+3*Sqrt[2])^n +(1-3*Sqrt[2])^n)/2, {n, 0, 30}]]
(* alternate program *)
LinearRecurrence[{2, 17}, {1, 1}, 30] (* T. D. Noe, Mar 28 2012 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-x)/(1-2*x-17*x^2)) \\ G. C. Greubel, Aug 03 2019
(Magma) I:=[1, 1]; [n le 2 select I[n] else 2*Self(n-1) +17*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 03 2019
(Sage) ((1-x)/(1-2*x-17*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019
(GAP) a:=[1, 1];; for n in [3..30] do a[n]:=2*a[n-1]+17*a[n-2]; od; a; # G. C. Greubel, Aug 03 2019

Crossrefs

Cf. A125817.

Sequence in context: A069131 A124712 A126373 * A362298 A367518 A093362

Adjacent sequences: A125815 A125816 A125817 * A125819 A125820 A125821

Keyword

nonn

Author

Artur Jasinski, Dec 10 2006

Status

approved