A133407 a(n) = a(n-1) + 5*a(n-2) for n >= 2, a(0)=1, a(1)=2.

1, 2, 7, 17, 52, 137, 397, 1082, 3067, 8477, 23812, 66197, 185257, 516242, 1442527, 4023737, 11236372, 31355057, 87536917, 244312202, 681996787, 1903557797, 5313541732, 14831330717, 41399039377, 115555692962, 322550889847, 900329354657, 2513083803892

Offset: 0

Author

Philippe Deléham, Jan 03 2008

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 100 terms from Matt C. Anderson)
• Index entries for linear recurrences with constant coefficients, signature (1,5).

Crossrefs

Cf. A030195 (shifted binomial transform).

Programs

Maple

a:= n-> (<<0|1>, <5|1>>^n. <<1, 2>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 20 2025

Mathematica

LinearRecurrence[{1,5},{1,2},30] (* Harvey P. Dale, Jul 23 2013 *)

PARI

x='x+O('x^99); Vec((1+x)/(1-x-5*x^2)) \\ Altug Alkan, Aug 28 2017

Formula

G.f.: (1+x)/(1-x-5*x^2).

a(n) = Sum_{k=0..n+1} A122950(n+1,k)*4^(n+1-k). - Philippe Deléham, Jan 08 2008

a(n) = ((21 - 3*sqrt(21))/42)*(1/2 - (1/2)*sqrt(21))^n + ((21 + 3*sqrt(21))/42)*(1/2 + (1/2)*sqrt(21))^n. - Richard Choulet, Nov 20 2008