A270704 Even 14-gonal (or tetradecagonal) numbers.

0, 14, 76, 186, 344, 550, 804, 1106, 1456, 1854, 2300, 2794, 3336, 3926, 4564, 5250, 5984, 6766, 7596, 8474, 9400, 10374, 11396, 12466, 13584, 14750, 15964, 17226, 18536, 19894, 21300, 22754, 24256, 25806, 27404, 29050, 30744, 32486, 34276, 36114, 38000

Offset

0,2

Comments

First bisection of A051866.

More generally, the ordinary generating function for the even k-gonal numbers with even k or for the first bisection of k-gonal numbers, is (k*x + (3*k - 8)*x^2)/(1 - x )^3.

Links

Table of n, a(n) for n=0..40.

OEIS Wiki, Figurate numbers

Index entries for linear recurrences with constant coefficients, signature (3,-3,1)

Formula

G.f.: 2*x*(7 + 17*x)/(1 - x)^3.

E.g.f.: 2*exp(x)*x*(7 + 12*x).

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

a(n) = 2*n*(12*n - 5).

a(n) = A005843(n)*A017605(n-1).

Sum_{n>=1} 1/a(n) = (Pi - sqrt(3)*Pi + sqrt(3)*log(27) + sqrt(3)*log(64) + log(1728) + 6*log(sqrt(3)-1) + 2*sqrt(3)*log(sqrt(3)-1) - 6*log(sqrt(3)+1) - 2*sqrt(3)*log(sqrt(3)+1))/(20 + 20*sqrt(3)) = 0.102542837854…

Mathematica

LinearRecurrence[{3, -3, 1}, {0, 14, 76}, 41]
Table[2 n (12 n - 5), {n, 0, 40}]
PolygonalNumber[14, Range[0, 80, 2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 12 2017 *)

Prog

(PARI) concat(0, Vec(2*x*(7 + 17*x)/(1 - x)^3 + O(x^60))) \\ Michel Marcus, Mar 22 2016

Crossrefs

Cf. similar sequences of the even k-gonal numbers with even k: A016742 (k = 4), A014635 (k = 6), A014642 (k = 8), A028994 (k = 10), A193872 (k = 12).

Sequence in context: A108650 A093567 A296996 * A200554 A152100 A173962

Adjacent sequences: A270701 A270702 A270703 * A270705 A270706 A270707

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Mar 22 2016

Status

approved