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

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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). [From Reinhard Zumkeller, Jul 13 2010]

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

Cf. A156771, A156773.

Sequence in context: A107814 A093762 A354085 * A097398 A225210 A033363

Adjacent sequences: A156674 A156675 A156676 * A156678 A156679 A156680

Keyword

nonn,easy

Author

Vincenzo Librandi, Feb 15 2009

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010

Status

approved