%I #41 Feb 13 2023 12:31:08
%S 0,11,34,69,116,175,246,329,424,531,650,781,924,1079,1246,1425,1616,
%T 1819,2034,2261,2500,2751,3014,3289,3576,3875,4186,4509,4844,5191,
%U 5550,5921,6304,6699,7106,7525,7956,8399,8854,9321,9800,10291,10794,11309,11836,12375
%N Second 14-gonal numbers: n*(6*n+5).
%C Sequence found by reading the line from 0, in the direction 0, 34, ... and the line from 11 in the direction 11, 69, ..., in the square spiral whose vertices are the generalized 14-gonal numbers A195818.
%H Ivan Panchenko, <a href="/A211014/b211014.txt">Table of n, a(n) for n = 0..1000</a>
%H Mark W. Coffey, <a href="https://arxiv.org/abs/1601.01673">Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers</a>, arXiv:1601.01673 [math.NT], 2016. See 3rd formula in Proposition 3 p. 36 giving a(n+1).
%H Leo Tavares, <a href="/A211014/a211014_1.jpg">Star illustration</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = -2*Sum_{k=0..n-1} binomial(6*n+5, 6*k+8)*Bernoulli(6*k+8)). - _Michel Marcus_, Jan 11 2016
%F From _G. C. Greubel_, Jul 04 2019: (Start)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F G.f.: x*(11+x)/(1-x)^3.
%F E.g.f.: x*(11+6*x)*exp(x). (End)
%F From _Amiram Eldar_, Feb 28 2022: (Start)
%F Sum_{n>=1} 1/a(n) = sqrt(3)*Pi/10 + 6/25 - 3*log(3)/10 - 2*log(2)/5.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/5 + log(2)/5 - 6/25 - sqrt(3)*log(sqrt(3)+2)/5. (End)
%F a(n) = A003154(n+1) - n - 1. - _Leo Tavares_, Jan 29 2023
%t Table[n*(6*n+5), {n,0,50}] (* _G. C. Greubel_, Jul 04 2019 *)
%t LinearRecurrence[{3,-3,1},{0,11,34},50] (* _Harvey P. Dale_, Dec 12 2022 *)
%o (PARI) a(n)=n*(6*n+5) \\ _Charles R Greathouse IV_, Jun 17 2017
%o (Magma) [n*(6*n+5): n in [0..50]]; // _G. C. Greubel_, Jul 04 2019
%o (Sage) [n*(6*n+5) for n in (0..50)] # _G. C. Greubel_, Jul 04 2019
%o (GAP) List([0..50], n-> n*(6*n+5)) # _G. C. Greubel_, Jul 04 2019
%Y Bisection of A195818.
%Y Second k-gonal numbers (k=5..14): A005449, A014105, A147875, A045944, A179986, A033954, A062728, A135705, A211013, this sequence.
%Y Cf. A051866.
%Y Cf. A003154.
%K nonn,easy
%O 0,2
%A _Omar E. Pol_, Aug 04 2012