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A199405
y-values in the solution to 13*x^2 - 12 = y^2.
2
1, 14, 25, 155, 274, 1691, 2989, 18446, 32605, 201215, 355666, 2194919, 3879721, 23942894, 42321265, 261176915, 461654194, 2849003171, 5035874869, 31077857966, 54932969365, 339007434455, 599226788146, 3698003921039, 6536561700241, 40339035696974
OFFSET
1,2
COMMENTS
When are both n+1 and 13*n+1 perfect squares? This problem gives the equation 13*x^2-12=y^2.
FORMULA
a(n+4) = 11*a(n+2)-a(n) with a(1)=1, a(2)=14, a(3)=25, a(4)=155.
G.f.: x*(1+x)*(1+13*x+x^2)/((1+3*x-x^2)*(1-3*x-x^2)). - Bruno Berselli, Nov 08 2011
a(n) = 2^(-1-n)*(2*(3-sqrt(13))^n+(-3-sqrt(13))^n*(-3+sqrt(13))-3*(-3+sqrt(13))^n-sqrt(13)*(-3+sqrt(13))^n+2*(3+sqrt(13))^n). - Colin Barker, Mar 27 2016
MATHEMATICA
LinearRecurrence[{0, 11, 0, -1}, {1, 14, 25, 155}, 50] (* Bruno Berselli, Nov 08 2011 *)
PROG
(Magma) m:=27; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x)*(1+13*x+x^2)/((1+3*x-x^2)*(1-3*x-x^2)))); // Bruno Berselli, Nov 08 2011
(PARI) Vec(x*(1+x)*(1+13*x+x^2)/((1+3*x-x^2)*(1-3*x-x^2)) + O(x^50)) \\ Colin Barker, Mar 27 2016
CROSSREFS
Cf. A199404.
Sequence in context: A174519 A079505 A039604 * A256573 A030786 A094163
KEYWORD
nonn,easy
AUTHOR
Sture Sjöstedt, Nov 05 2011
EXTENSIONS
More terms from Bruno Berselli, Nov 08 2011
STATUS
approved