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%I #23 Mar 01 2022 05:33:32
%S 0,1,25,28,76,81,153,160,256,265,385,396,540,553,721,736,928,945,1161,
%T 1180,1420,1441,1705,1728,2016,2041,2353,2380,2716,2745,3105,3136,
%U 3520,3553,3961,3996,4428,4465,4921,4960,5440,5481,5985,6028,6556,6601,7153,7200,7776,7825,8425,8476,9100,9153
%N Generalized 28-gonal (or icosioctagonal) numbers: m*(13*m - 12) with m = 0, +1, -1, +2, -2, +3, -3, ...
%C Partial sums of A317324. - _Omar E. Pol_, Jul 28 2018
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F G.f.: x*(1 + 24*x + x^2) / ((1 + x)^2*(1 - x)^3). - _Vincenzo Librandi_, Jun 23 2018
%F From _Amiram Eldar_, Mar 01 2022: (Start)
%F a(n) = (26*n*(n + 1) + 11*(2*n + 1)*(-1)^n - 11)/8.
%F a(n) = n*(13*n + 24)/4, if n is even, or (n + 1)*(13*n - 11)/4 otherwise.
%F Sum_{n>=1} 1/a(n) = 13/144 + Pi*cot(Pi/13)/12. (End)
%t With[{nn = 54, s = 28}, {0}~Join~Riffle[Array[PolygonalNumber[s, #] &, Ceiling[nn/2]], Array[PolygonalNumber[s, -#] &, Ceiling[nn/2]]]] (* _Michael De Vlieger_, Jun 14 2018 *)
%t CoefficientList[Series[x (1 + 24 x + x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 60}], x] (* _Vincenzo Librandi_, Jun 23 2018 *)
%o (Magma) I:=[0,1,25,28,76]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // _Vincenzo Librandi_, Jun 23 2018
%Y Cf. A161935, A303814, A303815, A317324.
%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), 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), A316724 (k=26), A316725 (k=27), this sequence (k=28), A303815 (k=29), A316729 (k=30).
%K nonn,easy
%O 0,3
%A _Omar E. Pol_, Jun 12 2018
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