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A316724 Generalized 26-gonal (or icosihexagonal) numbers: m*(12*m - 11) with m = 0, +1, -1, +2, -2, +3, -3, ... 29
0, 1, 23, 26, 70, 75, 141, 148, 236, 245, 355, 366, 498, 511, 665, 680, 856, 873, 1071, 1090, 1310, 1331, 1573, 1596, 1860, 1885, 2171, 2198, 2506, 2535, 2865, 2896, 3248, 3281, 3655, 3690, 4086, 4123, 4541, 4580, 5020, 5061, 5523, 5566, 6050, 6095, 6601, 6648, 7176, 7225, 7775 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

48*a(n) + 121 is a square. - Bruno Berselli, Jul 11 2018

Partial sums of A317322. - Omar E. Pol, Jul 28 2018

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

FORMULA

From Bruno Berselli, Jul 11 2018: (Start)

O.g.f.: x*(1 + 22*x + x^2)/((1 + x)^2*(1 - x)^3).

a(n) = a(-1-n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

a(n) = (12*n*(n + 1) + 5*(2*n + 1)*(-1)^n - 5)/4. Therefore:

a(n) = n*(6*n + 11)/2 for n even; otherwise, a(n) = (n + 1)*(6*n - 5)/2.

(2*n - 1)*a(n) + (2*n + 1)*a(n-1) - n*(12*n^2 - 11) = 0.

(End)

MATHEMATICA

Table[(12 n (n + 1) + 5 (2 n + 1) (-1)^n - 5)/4, {n, 0, 60}] (* Bruno Berselli, Jul 11 2018 *)

CoefficientList[ Series[-x (x^2 + 22x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)

LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 23, 26, 70}, 50] (* Robert G. Wilson v, Jul 28 2018 *)

nn=30; Sort[Table[n (12 n - 11), {n, -nn, nn}]] (* Vincenzo Librandi, Jul 29 2018 *)

PROG

(PARI) concat(0, Vec(x*(1 + 22*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ Colin Barker, Jul 12 2018

CROSSREFS

Cf. A255185, A317322.

Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), this sequence (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

Sequence in context: A111452 A147627 A025059 * A160774 A163142 A038772

Adjacent sequences:  A316721 A316722 A316723 * A316725 A316726 A316727

KEYWORD

nonn,easy

AUTHOR

Omar E. Pol, Jul 11 2018

STATUS

approved

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Last modified March 29 02:19 EDT 2020. Contains 333104 sequences. (Running on oeis4.)