A164600 a(n) = 18*a(n-1) - 79*a(n-2) for n > 1; a(0) = 1, a(1) = 17.

1, 17, 227, 2743, 31441, 349241, 3802499, 40854943, 434991553, 4602307457, 48477201539, 509007338599, 5332433173201, 55772217368297, 582637691946467, 6081473282940943, 63438141429166081, 661450156372654961

Offset: 0

Author

Klaus Brockhaus, Aug 17 2009

Keywords

• nonn

Comments

Binomial transform of A081185 without initial term 0. Ninth binomial transform of A164587.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 0..975 (terms 0..100 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (18, -79).

Crossrefs

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), this sequence (m=8).

Cf. A147960, A153593, A164587.

Programs

Magma

[ n le 2 select 16*n-15 else 18*Self(n-1)-79*Self(n-2): n in [1..18] ];

Maple

m:=30; S:=series( (1-x)/(1-18*x+79*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 12 2021

Mathematica

LinearRecurrence[{18,-79},{1,17},30] (* Harvey P. Dale, Oct 30 2013 *)

PARI

my(x='x+O('x^50)); Vec((1-x)/(1-18*x+79*x^2)) \\ G. C. Greubel, Aug 11 2017

Sage

[( (1-x)/(1-18*x+79*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

Formula

a(n) = ((1+4*sqrt(2))*(9+sqrt(2))^n + (1-4*sqrt(2))*(9-sqrt(2))^n)/2.

G.f.: (1-x)/(1-18*x+79*x^2).

E.g.f.: exp(9*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - G. C. Greubel, Aug 11 2017

From G. C. Greubel, Mar 12 2021: (Start)

a(n) = 2*A147960(n) + 8*A153593(n).

a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*8^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)