A269548 - OEIS

%I #27 Jan 05 2025 19:51:40

%S -1,-233,-22961,-2250073,-220484321,-21605213513,-2117090440081,

%T -207453257914553,-20328302185186241,-1991966160890337193,

%U -195192355465067858801,-19126858869415759825433,-1874236976847279395033761,-183656096872163964953483273,-17996423256495221286046327121

%N Expansion of (-7*x^2 + 134*x + 1)/(x^3 - 99*x^2 + 99*x - 1).

%C 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 b_k.

%H J. Mc Laughlin, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/48-1/McLaughlin.pdf">An identity motivated by an amazing identity of Ramanujan</a>, Fib. Q., 48 (No. 1, 2010), 34-38.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99,-99,1).

%F G.f.: (-7*x^2 + 134*x + 1)/(x^3 - 99*x^2 + 99*x - 1).

%F a(n) = 4/3 + ((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

%t CoefficientList[Series[(-7 x^2 + 134 x + 1)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[FullSimplify[4/3 + ((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 *)

%o (PARI) Vec((-7*x^2 + 134*x + 1)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

%o (Sage)

%o gf = (-7*x^2+134*x+1)/(x^3-99*x^2+99*x-1)

%o print(taylor(gf, x, 0, 20).list()) # _Bruno Berselli_, Mar 01 2016

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((-7*x^2+134*x+1)/(x^3-99*x^2+99*x-1))); // _Bruno Berselli_, Mar 01 2016

%Y Cf. A261004, A269549, A269550, A269551, A269552, A269553, A269554, A269555, A269556.

%K sign,easy

%O 0,2

%A _Michel Marcus_, Feb 29 2016