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A270704 Even 14-gonal (or tetradecagonal) numbers. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 23 07:06 EST 2018. Contains 299473 sequences. (Running on oeis4.)