A158462 a(n) = 36*n^2 - 6.

30, 138, 318, 570, 894, 1290, 1758, 2298, 2910, 3594, 4350, 5178, 6078, 7050, 8094, 9210, 10398, 11658, 12990, 14394, 15870, 17418, 19038, 20730, 22494, 24330, 26238, 28218, 30270, 32394, 34590, 36858, 39198, 41610, 44094, 46650, 49278, 51978, 54750, 57594, 60510

Offset: 1

Author

Vincenzo Librandi, Mar 19 2009

Keywords

• nonn
• easy

Comments

The identity (12*n^2 - 1)^2 - (36*n^2 - 6)*(2*n)^2 = 1 can be written as A158463(n)^2 - a(n)*A005843(n)^2 = 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A140811, A158463.

Programs

Magma

I:=[30, 138, 318]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 12 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {30, 138, 318}, 50] (* Vincenzo Librandi, Feb 12 2012 *)
36Range[50]^2-6 (* Harvey P. Dale, Jul 19 2025 *)

PARI

for(n=1, 40, print1(36*n^2-6", ")); \\ Vincenzo Librandi, Feb 12 2012

Formula

G.f.: 6*x*(5 + 8*x - x^2)/(1-x)^3. - Bruno Berselli, Aug 27 2011

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 12 2012

From Amiram Eldar, Mar 05 2023: (Start)

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

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

From Elmo R. Oliveira, Jan 16 2025: (Start)

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

a(n) = 6*A140811(n). (End)