A123008 Expansion of x*(1 + 3*x)/(1 - 2*x - 25*x^2).

1, 5, 35, 195, 1265, 7405, 46435, 277995, 1716865, 10383605, 63688835, 386967795, 2366156465, 14406507805, 87966927235, 536096549595, 3271366280065, 19945146300005, 121674449601635, 741977556703395, 4525816353447665

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,25).

Crossrefs

Cf. A002534, A123004.

Programs

Magma

[n le 2 select 5^(n-1) else 2*Self(n-1) + 25*Self(n-2): n in [1..31]]; // G. C. Greubel, Jul 13 2021

Mathematica

M:= {{0, 5}, {5, 2}}; v[1] = {1, 1}; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n, 30}]

Sage

[(5*i)^(n-2)*(3*chebyshev_U(n-2, -i/5) + 5*i*chebyshev_U(n-1, -i/5)) for n in (1..30)] # G. C. Greubel, Jul 13 2021

Formula

From Colin Barker, Oct 19 2012: (Start)

a(n) = 2*a(n-1) + 25*a(n-2) for n>2.

G.f.: x*(1+3*x)/(1-2*x-25*x^2). (End)

a(n) = (5*i)^(n-2)*(3*ChebyshevU(n-2, -i/5) + 5*i*ChebyshevU(n-1, -i/5)). - G. C. Greubel, Jul 13 2021

Extensions

Sequence edited by Joerg Arndt and Colin Barker, Oct 19 2012