A269556 Expansion of (-7*x^2 + 148*x - 5)/(x^3 - 99*x^2 + 99*x - 1).

5, 347, 33865, 3318287, 325158125, 31862177827, 3122168268785, 305940628162967, 29979059391701845, 2937641879758617707, 287858925156952833305, 28207237023501619046047, 2764021369378001713679165, 270845886962020666321511987, 26540132900908647297794495425, 2600662178402085414517539039527

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 s_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, A269555.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((-7*x^2+148*x-5)/(x^3-99*x^2+99*x-1))); // Bruno Berselli, Mar 02 2016

Mathematica

CoefficientList[Series[(-7 x^2 + 148 x - 5)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[Simplify[17/12 + (-(17 Sqrt[6] - 43)/(2 Sqrt[6] + 5)^(2 n) + (17 Sqrt[6] + 43) (2 Sqrt[6] + 5)^(2 n))/24], {n, 0, 20}] (* Bruno Berselli, Mar 02 2016 *)

PARI

Vec((-7*x^2 + 148*x - 5)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

Sage

gf = (-7*x^2+148*x-5)/(x^3-99*x^2+99*x-1)
print(taylor(gf, x, 0, 20).list()) # Bruno Berselli, Mar 02 2016

Formula

G.f.: (-7*x^2 + 148*x - 5)/(x^3 - 99*x^2 + 99*x - 1).

a(n) = 17/12 + (-(17*sqrt(6) - 43)/(2*sqrt(6) + 5)^(2*n) + (17*sqrt(6) + 43)*(2 sqrt(6) + 5)^(2*n))/24. - Bruno Berselli, Mar 02 2016