A269555 - OEIS

%I #29 Jan 05 2025 19:51:40

%S 7,439,42767,4190479,410623927,40236954119,3942810879487,

%T 386355229235359,37858869654185447,3709782870880938199,

%U 363520862476677757807,35621334739843539326639,3490527283642190176252567,342036052462194793733424679,33516042614011447595699365727,3284230140120659669584804416319

%N Expansion of (x^2 + 254*x - 7)/(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 r_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.: (x^2 + 254*x - 7)/(x^3 - 99*x^2 + 99*x - 1).

%F 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

%t 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 *)

%t LinearRecurrence[{99,-99,1},{7,439,42767},20] (* _Harvey P. Dale_, Apr 10 2019 *)

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

%o (Sage)

%o gf = (x^2+254*x-7)/(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!((x^2+254*x-7)/(x^3-99*x^2+99*x-1))); // _Bruno Berselli_, Mar 01 2016

%Y Cf. A261004, A269548, A269549, A269550, A269551, A269552, A269553, A269554, A269556.

%K nonn,easy,changed

%O 0,1

%A _Michel Marcus_, Feb 29 2016