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%I #56 Sep 08 2022 08:46:02
%S 1,6,21,46,81,126,181,246,321,406,501,606,721,846,981,1126,1281,1446,
%T 1621,1806,2001,2206,2421,2646,2881,3126,3381,3646,3921,4206,4501,
%U 4806,5121,5446,5781,6126,6481,6846,7221,7606,8001,8406,8821,9246,9681,10126,10581,11046,11521,12006,12501
%N a(n) = 5*n^2 + 1.
%C 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)).
%C These are pentagonal-star numbers. - _Mario Cortés_, Oct 26 2020
%D Benjamin Fine & Gerhard Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Boston: Birkhäuser, 2007, page 268.
%H Vincenzo Librandi, <a href="/A212656/b212656.txt">Table of n, a(n) for n = 0..1000</a>
%H F. Javier de Vega, <a href="https://arxiv.org/abs/2003.13378">An extension of Furstenberg's theorem of the infinitude of primes</a>, arXiv:2003.13378 [math.NT], 2020.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 5*n^2 + 1 = (1 + n*sqrt(-5))*(1 - n*sqrt(-5)).
%F G.f.: (1+3*x+6*x^2)/(1-x)^3. - _Bruno Berselli_, May 23 2012
%F a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - _Vincenzo Librandi_, Jul 10 2012
%F From _Amiram Eldar_, Jul 15 2020: (Start)
%F Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(5))*coth(Pi/sqrt(5)))/2.
%F Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(5))*csch(Pi/sqrt(5)))/2. (End)
%F a(n) = A005891(n-1) + 5*A000217(n). - _Mario Cortés_, Oct 26 2020
%F From _Amiram Eldar_, Feb 05 2021: (Start)
%F Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(5))*sinh(sqrt(2/5)*Pi).
%F Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(5))*csch(Pi/sqrt(5)).(End)
%F E.g.f.: exp(x)*(1 + 5*x + 5*x^2). - _Stefano Spezia_, Feb 05 2021
%t Table[5n^2 + 1, {n, 0, 49}]
%t LinearRecurrence[{3,-3,1},{1,6,21},60] (* _Harvey P. Dale_, Apr 04 2017 *)
%o (Magma) [5*n^2 + 1: n in [0..50]]; // _Vincenzo Librandi_, Jul 10 2012
%o (PARI) a(n)=5*n^2+1 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A033429, A134538, A002522, A053755, A056107, A058331.
%Y Cf. A137530 (primes of the form 1+5*n^2).
%K nonn,easy
%O 0,2
%A _Alonso del Arte_, May 23 2012
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