A269551 Expansion of (3*x^2 + 258*x - 5)/(x^3 - 99*x^2 + 99*x - 1).

5, 237, 22965, 2250077, 220484325, 21605213517, 2117090440085, 207453257914557, 20328302185186245, 1991966160890337197, 195192355465067858805, 19126858869415759825437, 1874236976847279395033765, 183656096872163964953483277, 17996423256495221286046327125, 1763465823039659522067586574717

Offset: 0

Author

Michel Marcus, Feb 29 2016

Keywords

• nonn
• easy

Comments

Mc Laughlin (2010) gives an identity relating ten sequences, denoted a_k, b_k, ..., f_k, p_k, q_k, r_k, s_k. This is the sequence e_k.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
• Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Crossrefs

Cf. A010701, A261004, A269548, A269549, A269550, A269552, A269553, A269554, A269555, A269556.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((3*x^2+258*x-5)/(x^3-99*x^2+99*x-1))); // Bruno Berselli, Mar 01 2016

Mathematica

CoefficientList[Series[(3 x^2 + 258 x - 5)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[FullSimplify[8/3 + (-(3 Sqrt[6] - 7)/(2 Sqrt[6] + 5)^(2 n) + (3 Sqrt[6] + 7) (2 Sqrt[6] + 5)^(2 n))/6], {n, 0, 20}] (* Bruno Berselli, Mar 01 2016 *)

PARI

Vec((3*x^2 + 258*x - 5)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

Sage

gf = (3*x^2+258*x-5)/(x^3-99*x^2+99*x-1)
print(taylor(gf, x, 0, 20).list()) # Bruno Berselli, Mar 01 2016

Formula

G.f.: (3*x^2 + 258*x - 5)/(x^3 - 99*x^2 + 99*x - 1).

a(n) = 8/3 + (-(3*sqrt(6) - 7)/(2*sqrt(6) + 5)^(2*n) + (3*sqrt(6) + 7)*(2*sqrt(6) + 5)^(2*n))/6. - Bruno Berselli, Mar 01 2016