A158480 a(n) = 12*n^2 + 1.

1, 13, 49, 109, 193, 301, 433, 589, 769, 973, 1201, 1453, 1729, 2029, 2353, 2701, 3073, 3469, 3889, 4333, 4801, 5293, 5809, 6349, 6913, 7501, 8113, 8749, 9409, 10093, 10801, 11533, 12289, 13069, 13873, 14701, 15553, 16429, 17329, 18253, 19201, 20173, 21169

Offset: 0

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 a(n)^2 - A158479(n)*A005843(n)^2 = 1.

Sequence found by reading the line from 13, in the direction 13, 49, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

Examples

			a(1) = 12*1^2 + 1 = 13.
a(2) = 12*2^2 + 1 = 49.
a(3) = 12*3^2 + 1 = 109.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Vincenzo Librandi, X^2-AY^2=1 [broken link]
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A158479.

Programs

GAP

List([1..40], n-> 12*n^2 + 1); # G. C. Greubel, Nov 06 2019

Magma

I:=[13,49,109];[n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2) +Self(n-3): n in [1..50]];

Maple

seq(12*n^2 +1, n=0..45); # G. C. Greubel, Nov 06 2019

Mathematica

LinearRecurrence[{3,-3,1}, {13,49,109}, 40]
12*Range[40]^2 +1 (* G. C. Greubel, Nov 06 2019 *)

PARI

a(n)=12*n^2+1

Sage

[12*n^2 +1 for n in (1..40)] # G. C. Greubel, Nov 06 2019

Formula

a(n) = A010014(n)/2. - Vladimir Joseph Stephan Orlovsky, May 18 2009

G.f: (13*x^2 + 10*x + 1)/(1-x)^3.

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

a(n) = 1 + A135453(n). - Omar E. Pol, Jul 18 2012

a(n) = (A016969(n-1)*A016921(n) + 4)/3. - Hilko Koning, Oct 25 2019

E.g.f.: (1 + 12*x + 12*x^2)*exp(x). - G. C. Greubel, Nov 06 2019

From Amiram Eldar, Jul 15 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(12))*coth(Pi/sqrt(12)))/2.

Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(12))*csch(Pi/sqrt(12)))/2. (End)

From Amiram Eldar, Feb 05 2021: (Start)

Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(12))*sinh(Pi/sqrt(6)).

Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(12))*csch(Pi/sqrt(12)). (End)

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 12 2021