This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A211014 #48 Feb 26 2025 19:22:25
%S A211014 0,11,34,69,116,175,246,329,424,531,650,781,924,1079,1246,1425,1616,
%T A211014 1819,2034,2261,2500,2751,3014,3289,3576,3875,4186,4509,4844,5191,
%U A211014 5550,5921,6304,6699,7106,7525,7956,8399,8854,9321,9800,10291,10794,11309,11836,12375
%N A211014 Second 14-gonal numbers: n*(6*n+5).
%C A211014 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 A211014 Ivan Panchenko, <a href="/A211014/b211014.txt">Table of n, a(n) for n = 0..1000</a>
%H A211014 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 A211014 Leo Tavares, <a href="/A211014/a211014_1.jpg">Star illustration</a>
%H A211014 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A211014 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 A211014 From _G. C. Greubel_, Jul 04 2019: (Start)
%F A211014 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A211014 G.f.: x*(11+x)/(1-x)^3.
%F A211014 E.g.f.: x*(11+6*x)*exp(x). (End)
%F A211014 From _Amiram Eldar_, Feb 28 2022: (Start)
%F A211014 Sum_{n>=1} 1/a(n) = sqrt(3)*Pi/10 + 6/25 - 3*log(3)/10 - 2*log(2)/5.
%F A211014 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/5 + log(2)/5 - 6/25 - sqrt(3)*log(sqrt(3)+2)/5. (End)
%F A211014 a(n) = A003154(n+1) - n - 1. - _Leo Tavares_, Jan 29 2023
%t A211014 Table[n*(6*n+5), {n,0,50}] (* _G. C. Greubel_, Jul 04 2019 *)
%t A211014 LinearRecurrence[{3,-3,1},{0,11,34},50] (* _Harvey P. Dale_, Dec 12 2022 *)
%o A211014 (PARI) a(n)=n*(6*n+5) \\ _Charles R Greathouse IV_, Jun 17 2017
%o A211014 (Magma) [n*(6*n+5): n in [0..50]]; // _G. C. Greubel_, Jul 04 2019
%o A211014 (Sage) [n*(6*n+5) for n in (0..50)] # _G. C. Greubel_, Jul 04 2019
%o A211014 (GAP) List([0..50], n-> n*(6*n+5)); # _G. C. Greubel_, Jul 04 2019
%Y A211014 Bisection of A195818.
%Y A211014 Second k-gonal numbers (k=5..14): A005449, A014105, A147875, A045944, A179986, A033954, A062728, A135705, A211013, this sequence.
%Y A211014 Cf. A051866.
%Y A211014 Cf. A003154.
%K A211014 nonn,easy
%O A211014 0,2
%A A211014 _Omar E. Pol_, Aug 04 2012