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

Author

Ilya Gutkovskiy, Mar 22 2016

Keywords

• nonn
• easy

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

• OEIS Wiki, Figurate numbers
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1)

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).

Programs

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 *)

PARI

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

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…