A122995 Expansion of x*(1+4*x)/(1-x-25*x^2).

1, 5, 30, 155, 905, 4780, 27405, 146905, 832030, 4504655, 25305405, 137921780, 770556905, 4218601405, 23482524030, 128947559155, 716010659905, 3939699638780, 21839966136405, 120332457105905, 666331610516030, 3674643038163655, 20332933301064405, 112199009255155780

Offset: 1

Author

Roger L. Bagula, Sep 22 2006

Keywords

• nonn
• easy

Comments

The sequence can also be generated by adding of the top-row elements of the (n-1)st power of the matrix [[0,1],[1,1/5]] and multiplying with 5^(n-1).

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (1,25)

Crossrefs

Cf. A026597.

Programs

Mathematica

M := {{0, 1}, {1, 1/5}}; v[1] = {1, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[v[n][[1]]*5^(n - 1), {n, 1, 30}]
LinearRecurrence[{1,25},{1,5},30] (* Harvey P. Dale, Mar 11 2017 *)

PARI

Vec(x*(1+4*x)/(1-x-25*x^2) + O(x^30)) \\ Michel Marcus, Jan 28 2015

Formula

a(n) = a(n-1)+25*a(n-2). [Philippe Deléham, Mar 26 2009]

a(n) = A122999(n-1) + 4*A122999(n-2). [R. J. Mathar, Aug 12 2009]

a(n) = (1/2+9*sqrt(101)/202)*(1/2+sqrt(101)/2)^(n-1) + (1/2-9*sqrt(101)/202)*(1/2-sqrt(101)/2)^(n-1). [Antonio Alberto Olivares, Jun 07 2011]

Extensions

Replaced definition by a specific one - The Assoc. Eds. of the OEIS, Jun 07 2010

More terms from Michel Marcus, Jan 28 2015