A158675 a(n) = 961*n^2 + 31.

31, 992, 3875, 8680, 15407, 24056, 34627, 47120, 61535, 77872, 96131, 116312, 138415, 162440, 188387, 216256, 246047, 277760, 311395, 346952, 384431, 423832, 465155, 508400, 553567, 600656, 649667, 700600, 753455, 808232, 864931, 923552, 984095, 1046560, 1110947

Offset

0,1

Comments

The identity (62*n^2 + 1)^2 - (961*n^2 + 31)*(2*n)^2 = 1 can be written as A158676(n)^2 - a(n)*A005843(n)^2 = 1.

Links

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

G.f.: -31*(1 + 29*x + 32*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 21 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(31))*Pi/sqrt(31) + 1)/62.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(31))*Pi/sqrt(31) + 1)/62. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {31, 992, 3875}, 50] (* Vincenzo Librandi, Feb 19 2012 *)
961 Range[0, 40]^2+31 (* or *) CoefficientList[Series[-((31 (1+29 x+32 x^2))/(-1+x)^3), {x, 0, 40}], x] (* Harvey P. Dale, Jul 31 2021 *)

Prog

(Magma) I:=[31, 992, 3875]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 19 2012
(PARI) for(n=0, 40, print1(961*n^2 + 31", ")); \\ Vincenzo Librandi, Feb 19 2012

Crossrefs

Cf. A005843, A158676.

Sequence in context: A042862 A159674 A138958 * A154808 A307588 A218285

Adjacent sequences: A158672 A158673 A158674 * A158676 A158677 A158678

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 24 2009

Extensions

Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved