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A157461
Expansion of x*(x+1) / (x^2-26*x+1).
4
1, 27, 701, 18199, 472473, 12266099, 318446101, 8267332527, 214632199601, 5572169857099, 144661784084973, 3755634216352199, 97501827841072201, 2531291889651525027, 65716087303098578501, 1706086977990911515999, 44292545340460600837473
OFFSET
1,2
COMMENTS
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:
A * c(n)+1 = a(n)^2,
(A+1) * c(n)+1 = b(n)^2, for details see comment in A157014.
A157461 is the b(n) sequence for A=6.
Numbers k such that 42*k^2 + 7 is a square. - Klaus Purath, Jun 12 2021
FORMULA
G.f.: x*(x+1) / (x^2-26*x+1).
a(1) = 1, a(2) = 27, a(n) = 26*a(n-1)-a(n-2) for n>2.
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
a(n+1) = (a(n)^2 - 28)/a(n-1), n > 1. - Klaus Purath, Jun 12 2021
PROG
(PARI) Vec(x*(x+1)/(x^2-26*x+1)+O(x^20)) \\ Charles R Greathouse IV, Sep 26 2012
(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
CROSSREFS
6*A157874(n)+1 = A153111(n)^2.
7*A157874(n)+1 = A157461(n)^2.
Sequence in context: A113364 A095898 A014914 * A342037 A162827 A163179
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, Mar 01 2009
EXTENSIONS
Edited by Alois P. Heinz, Sep 09 2011
STATUS
approved