A158479 a(n) = 36*n^2 + 6.

42, 150, 330, 582, 906, 1302, 1770, 2310, 2922, 3606, 4362, 5190, 6090, 7062, 8106, 9222, 10410, 11670, 13002, 14406, 15882, 17430, 19050, 20742, 22506, 24342, 26250, 28230, 30282, 32406, 34602, 36870, 39210, 41622, 44106, 46662, 49290, 51990, 54762, 57606, 60522

Offset: 1

Author

Vincenzo Librandi, Mar 20 2009

Keywords

• nonn
• easy

Comments

The identity (12*n^2 + 1)^2 - (36*n^2 + 6)*(2*n)^2 = 1 can be written as A158480(n)^2 - a(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, A158480, A227776.

Programs

Magma

I:=[42, 150, 330]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

Mathematica

LinearRecurrence[{3,-3,1},{42,150,330},40]

PARI

a(n)=36*n^2+6 \\ Charles R Greathouse IV, Jun 17 2017

Formula

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

G.f: 6*x*(7+4*x+x^2)/(1-x)^3.

From Amiram Eldar, Mar 05 2023: (Start)

Sum_{n>=1} 1/a(n) = (coth(Pi/sqrt(6))*Pi/sqrt(6) - 1)/12.

Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/sqrt(6))*Pi/sqrt(6))/12. (End)

From Elmo R. Oliveira, Jan 15 2025: (Start)

E.g.f.: 6*(exp(x)*(6*x^2 + 6*x + 1) - 1).

a(n) = 6*A227776(n). (End)