A269555 Expansion of (x^2 + 254*x - 7)/(x^3 - 99*x^2 + 99*x - 1).

7, 439, 42767, 4190479, 410623927, 40236954119, 3942810879487, 386355229235359, 37858869654185447, 3709782870880938199, 363520862476677757807, 35621334739843539326639, 3490527283642190176252567, 342036052462194793733424679, 33516042614011447595699365727, 3284230140120659669584804416319

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 r_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, A269551, A269552, A269553, A269554, A269556.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^2+254*x-7)/(x^3-99*x^2+99*x-1))); // Bruno Berselli, Mar 01 2016

Mathematica

CoefficientList[Series[(x^2 + 254 x - 7)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[FullSimplify[31/12 + (-(22 Sqrt[6] - 53)/(2 Sqrt[6] + 5)^(2 n) + (22 Sqrt[6] + 53) (2 Sqrt[6] + 5)^(2 n))/24], {n, 0, 20}] (* Bruno Berselli, Mar 01 2016 *)
LinearRecurrence[{99,-99,1},{7,439,42767},20] (* Harvey P. Dale, Apr 10 2019 *)

PARI

Vec((x^2 + 254*x - 7)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

Sage

gf = (x^2+254*x-7)/(x^3-99*x^2+99*x-1)
print(taylor(gf, x, 0, 20).list()) # Bruno Berselli, Mar 01 2016

Formula

G.f.: (x^2 + 254*x - 7)/(x^3 - 99*x^2 + 99*x - 1).

a(n) = 31/12 + (-(22*sqrt(6) - 53)/(2*sqrt(6) + 5)^(2*n) + (22*sqrt(6) + 53)*(2*sqrt(6)+5)^(2*n))/24. - Bruno Berselli, Mar 01 2016