A157889 a(n) = 18*n^2 + 1.

19, 73, 163, 289, 451, 649, 883, 1153, 1459, 1801, 2179, 2593, 3043, 3529, 4051, 4609, 5203, 5833, 6499, 7201, 7939, 8713, 9523, 10369, 11251, 12169, 13123, 14113, 15139, 16201, 17299, 18433, 19603, 20809, 22051, 23329, 24643, 25993, 27379, 28801, 30259, 31753

Offset

1,1

Comments

The identity (18n^2 + 1)^2 - (81n^2 + 9)*(2n)^2 = 1 can be written as a(n)^2 - A157888(n)*A005843(n+1)^2 = 1. - Vincenzo Librandi, Feb 05 2012

Sequence found by reading the line from 19, in the direction 19, 73, ... in the square spiral whose vertices are the generalized hendecagonal numbers A195160. - Omar E. Pol, Nov 05 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]

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

Formula

From Vincenzo Librandi, Feb 05 2012: (Start)

G.f: x*(19 + 16*x + x^2)/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

From Amiram Eldar, Mar 07 2023: (Start)

Sum_{n>=1} 1/a(n) = (coth(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)) - 1)/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)))/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {19, 73, 163}, 40] (* Vincenzo Librandi, Feb 05 2012 *)

Prog

(Magma) I:=[19, 73, 163]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 05 2012
(PARI) for(n=1, 40, print1(18n^2 + 1", ")); \\ Vincenzo Librandi, Feb 05 2012

Crossrefs

Cf. A005843, A157888, A195160.

Sequence in context: A154406 A141960 A178541 * A361677 A255897 A220447

Adjacent sequences: A157886 A157887 A157888 * A157890 A157891 A157892

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 08 2009

Status

approved