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A211014
Second 14-gonal numbers: n*(6*n+5).
6
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
0,2
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
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
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
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 *)
PROG
(PARI) a(n)=n*(6*n+5) \\ Charles R Greathouse IV, Jun 17 2017
(Magma) [n*(6*n+5): n in [0..50]]; // G. C. Greubel, Jul 04 2019
(Sage) [n*(6*n+5) for n in (0..50)] # G. C. Greubel, Jul 04 2019
(GAP) List([0..50], n-> n*(6*n+5)) # G. C. Greubel, Jul 04 2019
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.
Sequence in context: A226502 A041230 A063162 * A103661 A300418 A041539
KEYWORD
nonn,easy,changed
AUTHOR
Omar E. Pol, Aug 04 2012
STATUS
approved