A123011 a(n) = 2*a(n-1) + 5*a(n-2) for n > 1; a(0) = 1, a(1) = 5.

1, 5, 15, 55, 185, 645, 2215, 7655, 26385, 91045, 314015, 1083255, 3736585, 12889445, 44461815, 153370855, 529050785, 1824955845, 6295165615, 21715110455, 74906048985, 258387650245, 891305545415, 3074549342055

Offset: 0

Author

Roger L. Bagula, Sep 23 2006

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,5).

Crossrefs

Cf. A002532, A164532, A164549.

Programs

Magma

[ n le 2 select 4*n-3 else 2*Self(n-1)+5*Self(n-2): n in [1..24] ]; // Klaus Brockhaus, Aug 15 2009

Mathematica

LinearRecurrence[{2,5}, {1,5}, 31] (* G. C. Greubel, Jul 13 2021 *)

Sage

[1]+[(sqrt(5)*i)^(n-1)*(sqrt(5)*i*chebyshev_U(n, -i/sqrt(5)) + 3*chebyshev_U(n-1, -i/sqrt(5))) for n in (1..30)] # G. C. Greubel, Jul 13 2021

Formula

a(n) = ((3+2*sqrt(6))*(1+sqrt(6))^n + (3-2*sqrt(6))*(1-sqrt(6))^n)/6. - Klaus Brockhaus, Aug 15 2009

From Klaus Brockhaus, Aug 15 2009: (Start)

G.f.: (1+3*x)/(1-2*x-5*x^2).

Binomial transform of A164532.

Inverse binomial transform of A164549. (End)

a(n) = (sqrt(5)*i)^(n-1)*(sqrt(5)*i*ChebyshevU(n, -i/sqrt(5)) + 3*ChebyshevU(n-1, -i/sqrt(5))) for n > 0 with a(0) = 1. - G. C. Greubel, Jul 13 2021

Extensions

Edited by N. J. A. Sloane, Aug 27 2009, using simpler definition suggested by Klaus Brockhaus, Aug 15 2009