A156677 a(n) = 81*n^2 - 118*n + 43.

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

Author

Vincenzo Librandi, Feb 15 2009

Keywords

• nonn
• easy

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A017305, A156771, A156773, A171198.

Programs

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]];

Mathematica

LinearRecurrence[{3,-3,1},{43,6,131},40]

PARI

a(n)=81*n^2-118*n+43 \\ Charles R Greathouse IV, Dec 23 2011

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

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010