OFFSET
0,2
COMMENTS
Solutions to X*(X+1)=42*Y^2 with Y=A097309. - R. J. Mathar, Mar 14 2023
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (27,-27,1).
FORMULA
sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(7) + sqrt(6))^n.
sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(7) - sqrt(6))^n.
a(n) = 27*a(n-1) - 27*a(n-2) + a(n-3) for n > 2.
From Colin Barker, Dec 24 2018: (Start)
G.f.: 6*x*(1 + x) / ((1 - x)*(1 - 26*x + x^2)).
a(n) = ((13+2*sqrt(42))^(-n) * (-1+(13+2*sqrt(42))^n)^2) / 4.
(End)
2*a(n) = A097308(n)-1. - R. J. Mathar, Mar 14 2023
EXAMPLE
(sqrt(7) + sqrt(6))^2 = 13 + 2*sqrt(42) = sqrt(169) + sqrt(168). So a(2) = 168.
MATHEMATICA
LinearRecurrence[{27, -27, 1}, {0, 6, 168}, 20] (* Harvey P. Dale, Apr 30 2022 *)
PROG
(PARI) concat(0, Vec(6*x*(1 + x) / ((1 - x)*(1 - 26*x + x^2)) + O(x^20))) \\ Colin Barker, Dec 24 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Dec 24 2018
STATUS
approved