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A156677
a(n) = 81*n^2 - 118*n + 43.
5
43, 6, 131, 418, 867, 1478, 2251, 3186, 4283, 5542, 6963, 8546, 10291, 12198, 14267, 16498, 18891, 21446, 24163, 27042, 30083, 33286, 36651, 40178, 43867, 47718, 51731, 55906, 60243, 64742, 69403, 74226, 79211, 84358, 89667, 95138, 100771, 106566, 112523, 118642
OFFSET
0,1
COMMENTS
The identity (6561*n^2 - 9558*n + 3482)^2 - (81*n^2 - 118*n + 43)*(729*n - 531)^2 = 1 can be written as A156773(n)^2 - a(n)*A156771(n)^2 = 1 for n > 0.
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [9n-7; {2, 4, 9n-7, 4, 2, 18n-14}]. For n=1, this collapses to [2; {2, 4}]. - Magus K. Chu, Sep 09 2022
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (-43+123*x-242*x^2)/(x-1)^3.
For n > 1: a(n) = A171198(n-2) - A017305(n-2). - Reinhard Zumkeller, Jul 13 2010
E.g.f.: exp(x)*(43 - 37*x + 81*x^2). - Elmo R. Oliveira, Oct 19 2024
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {43, 6, 131}, 40]
PROG
(Magma) I:=[43, 6, 131]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
(PARI) a(n)=81*n^2-118*n+43 \\ Charles R Greathouse IV, Dec 23 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Feb 15 2009
EXTENSIONS
Edited by Charles R Greathouse IV, Jul 25 2010
STATUS
approved