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A303305 Generalized 17-gonal (or heptadecagonal) numbers: m*(15*m - 13)/2 with m = 0, +1, -1, +2, -2, +3, -3, ... 30

%I #49 May 01 2021 21:46:19

%S 0,1,14,17,43,48,87,94,146,155,220,231,309,322,413,428,532,549,666,

%T 685,815,836,979,1002,1158,1183,1352,1379,1561,1590,1785,1816,2024,

%U 2057,2278,2313,2547,2584,2831,2870,3130,3171,3444,3487,3773,3818,4117,4164,4476,4525,4850

%N Generalized 17-gonal (or heptadecagonal) numbers: m*(15*m - 13)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

%C 120*a(n) + 169 is a square. - _Bruno Berselli_, Jun 08 2018

%C Partial sums of A317313. - _Omar E. Pol_, Jul 28 2018

%C Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. They are also the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, k >= 5. In this case k = 17. - _Omar E. Pol_, Apr 25 2021

%H Colin Barker, <a href="/A303305/b303305.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F From _Bruno Berselli_, Jun 08 2018: (Start)

%F G.f.: x*(1 + 13*x + x^2)/((1 + x)^2*(1 - x)^3).

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

%F a(n) = (30*n*(n + 1) + 11*(2*n + 1)*(-1)^n - 11)/16. Therefore:

%F a(n) = n*(15*n + 26)/8, if n is even, or (n + 1)*(15*n - 11)/8 otherwise.

%F 2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) - n*(15*n^2 - 13) = 0. (End)

%e From _Omar E. Pol_, Apr 24 2021: (Start)

%e Illustration of initial terms as vertices of a rectangular spiral:

%e 43_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _17

%e | |

%e | 0 |

%e | |_ _ _ _ _ _ _ _ _ _ _ _ _|

%e | 1 14

%e |

%e 48

%e More generally, all generalized k-gonal numbers can be represented with this kind of spirals, k >= 5". (End)

%t With[{pp = 17, nn = 55}, {0}~Join~Riffle[Array[PolygonalNumber[pp, #] &, Ceiling[nn/2]], Array[PolygonalNumber[pp, -#] &, Ceiling[nn/2]]]] (* _Michael De Vlieger_, Jun 06 2018 *)

%t Table[(30 n (n + 1) + 11 (2 n + 1) (-1)^n - 11)/16, {n, 0, 60}] (* _Bruno Berselli_, Jun 08 2018 *)

%t CoefficientList[ Series[-x (x^2 + 13x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)

%t LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 14, 17, 43}, 51] (* _Robert G. Wilson v_, Jul 28 2018 *)

%o (PARI) concat(0, Vec(x*(1 + 13*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ _Colin Barker_, Jun 12 2018

%Y Cf. A051869, A194715, A244636, A317313.

%Y 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), this sequence (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), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Jun 06 2018

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