A212656 a(n) = 5*n^2 + 1.

1, 6, 21, 46, 81, 126, 181, 246, 321, 406, 501, 606, 721, 846, 981, 1126, 1281, 1446, 1621, 1806, 2001, 2206, 2421, 2646, 2881, 3126, 3381, 3646, 3921, 4206, 4501, 4806, 5121, 5446, 5781, 6126, 6481, 6846, 7221, 7606, 8001, 8406, 8821, 9246, 9681, 10126, 10581, 11046, 11521, 12006, 12501

Offset: 0

Author

Alonso del Arte, May 23 2012

Keywords

• nonn
• easy

Comments

Z[sqrt(-5)] is not a unique factorization domain, and some of the numbers in this sequence have two different factorizations in that domain, e.g., 21 = 3 * 7 = (1 + 2*sqrt(-5))*(1 - 2*sqrt(-5)). And of course some primes in Z are composite in Z[sqrt(-5)], like 181 = (1 + 6*sqrt(-5))*(1 - 6*sqrt(-5)).

These are pentagonal-star numbers. - Mario Cortés, Oct 26 2020

References

• Benjamin Fine & Gerhard Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Boston: Birkhäuser, 2007, page 268.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A033429, A134538, A002522, A053755, A056107, A058331.

Cf. A137530 (primes of the form 1+5*n^2).

Programs

Magma

[5*n^2 + 1: n in [0..50]]; // Vincenzo Librandi, Jul 10 2012

Mathematica

Table[5n^2 + 1, {n, 0, 49}]
LinearRecurrence[{3,-3,1},{1,6,21},60] (* Harvey P. Dale, Apr 04 2017 *)

PARI

a(n)=5*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 5*n^2 + 1 = (1 + n*sqrt(-5))*(1 - n*sqrt(-5)).

G.f.: (1+3*x+6*x^2)/(1-x)^3. - Bruno Berselli, May 23 2012

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Jul 10 2012

From Amiram Eldar, Jul 15 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(5))*coth(Pi/sqrt(5)))/2.

Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(5))*csch(Pi/sqrt(5)))/2. (End)

a(n) = A005891(n-1) + 5*A000217(n). - Mario Cortés, Oct 26 2020

From Amiram Eldar, Feb 05 2021: (Start)

Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(5))*sinh(sqrt(2/5)*Pi).

Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(5))*csch(Pi/sqrt(5)).(End)

E.g.f.: exp(x)*(1 + 5*x + 5*x^2). - Stefano Spezia, Feb 05 2021