A158738 a(n) = 72*n^2 - 1.

71, 287, 647, 1151, 1799, 2591, 3527, 4607, 5831, 7199, 8711, 10367, 12167, 14111, 16199, 18431, 20807, 23327, 25991, 28799, 31751, 34847, 38087, 41471, 44999, 48671, 52487, 56447, 60551, 64799, 69191, 73727, 78407, 83231, 88199, 93311, 98567, 103967, 109511, 115199

Offset: 1

Author

Vincenzo Librandi, Mar 25 2009

Keywords

• nonn
• easy

Comments

The identity (72*n^2 - 1)^2 - (1296*n^2 - 36)*(2*n)^2 = 1 can be written as a(n)^2 - A158737(n)*A005843(n)^2 = 1.

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).

Crossrefs

Cf. A005843, A157910, A158737.

Programs

Magma

I:=[71,287,647]; [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 21 2012

Maple

A158738:=n->72*n^2 - 1; seq(A158738(n), n=1..40); # Wesley Ivan Hurt, Feb 01 2014

Mathematica

72Range[40]^2-1 (* or *) LinearRecurrence[{3,-3,1},{71,287,647},40] (* Harvey P. Dale, May 01 2011 *)

PARI

for(n=1, 40, print1(72*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 21 2012

Formula

G.f.: x*(-71 - 74*x + x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 22 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(6*sqrt(2)))*Pi/(6*sqrt(2)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(6*sqrt(2)))*Pi/(6*sqrt(2)) - 1)/2. (End)

From Elmo R. Oliveira, Jan 17 2025: (Start)

E.g.f.: exp(x)*(72*x^2 + 72*x - 1) + 1.

a(n) = A157910(2*n). (End)

Extensions

Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009