A211014 Second 14-gonal numbers: n*(6*n+5).

0, 11, 34, 69, 116, 175, 246, 329, 424, 531, 650, 781, 924, 1079, 1246, 1425, 1616, 1819, 2034, 2261, 2500, 2751, 3014, 3289, 3576, 3875, 4186, 4509, 4844, 5191, 5550, 5921, 6304, 6699, 7106, 7525, 7956, 8399, 8854, 9321, 9800, 10291, 10794, 11309, 11836, 12375

Offset: 0

Author

Omar E. Pol, Aug 04 2012

Keywords

• nonn
• easy

Comments

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.

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, arXiv:1601.01673 [math.NT], 2016. See 3rd formula in Proposition 3 p. 36 giving a(n+1).
• Leo Tavares, Star illustration
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Bisection of A195818.

Second k-gonal numbers (k=5..14): A005449, A014105, A147875, A045944, A179986, A033954, A062728, A135705, A211013, this sequence.

Cf. A051866.

Cf. A003154.

Programs

GAP

List([0..50], n-> n*(6*n+5)); # G. C. Greubel, Jul 04 2019

Magma

[n*(6*n+5): n in [0..50]]; // G. C. Greubel, Jul 04 2019

Mathematica

Table[n*(6*n+5), {n,0,50}] (* G. C. Greubel, Jul 04 2019 *)
LinearRecurrence[{3,-3,1},{0,11,34},50] (* Harvey P. Dale, Dec 12 2022 *)

PARI

a(n)=n*(6*n+5) \\ Charles R Greathouse IV, Jun 17 2017

Sage

[n*(6*n+5) for n in (0..50)] # G. C. Greubel, Jul 04 2019

Formula

a(n) = -2*Sum_{k=0..n-1} binomial(6*n+5, 6*k+8)*Bernoulli(6*k+8). - Michel Marcus, Jan 11 2016

From G. C. Greubel, Jul 04 2019: (Start)

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

G.f.: x*(11+x)/(1-x)^3.

E.g.f.: x*(11+6*x)*exp(x). (End)

From Amiram Eldar, Feb 28 2022: (Start)

Sum_{n>=1} 1/a(n) = sqrt(3)*Pi/10 + 6/25 - 3*log(3)/10 - 2*log(2)/5.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/5 + log(2)/5 - 6/25 - sqrt(3)*log(sqrt(3)+2)/5. (End)

a(n) = A003154(n+1) - n - 1. - Leo Tavares, Jan 29 2023