A157461 - OEIS

%I #16 Jun 27 2021 03:29:02

%S 1,27,701,18199,472473,12266099,318446101,8267332527,214632199601,

%T 5572169857099,144661784084973,3755634216352199,97501827841072201,

%U 2531291889651525027,65716087303098578501,1706086977990911515999,44292545340460600837473

%N Expansion of x*(x+1) / (x^2-26*x+1).

%C This sequence is part of a solution of a more general problem involving two equations, three sequences a(n), b(n), c(n) and a constant A:

%C     A    * c(n)+1 = a(n)^2,

%C    (A+1) * c(n)+1 = b(n)^2, for details see comment in A157014.

%C A157461 is the b(n) sequence for A=6.

%C Numbers k such that 42*k^2 + 7 is a square. - _Klaus Purath_, Jun 12 2021

%H Colin Barker, <a href="/A157461/b157461.txt">Table of n, a(n) for n = 1..700</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (26,-1).

%F G.f.: x*(x+1) / (x^2-26*x+1).

%F a(1) = 1, a(2) = 27, a(n) = 26*a(n-1)-a(n-2) for n>2.

%F a(n) = (13+2*sqrt(42))^(-n)*(-6-sqrt(42)+(-6+sqrt(42))*(13+2*sqrt(42))^(2*n))/12. - _Colin Barker_, Jul 25 2016

%F a(n+1) = (a(n)^2 - 28)/a(n-1), n > 1. - _Klaus Purath_, Jun 12 2021

%o (PARI) Vec(x*(x+1)/(x^2-26*x+1)+O(x^20)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (PARI) a(n) = round((13+2*sqrt(42))^(-n)*(-6-sqrt(42)+(-6+sqrt(42))*(13+2*sqrt(42))^(2*n))/12) \\ _Colin Barker_, Jul 25 2016

%Y 6*A157874(n)+1 = A153111(n)^2.

%Y 7*A157874(n)+1 = A157461(n)^2.

%K nonn,easy

%O 1,2

%A _Paul Weisenhorn_, Mar 01 2009

%E Edited by _Alois P. Heinz_, Sep 09 2011