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Generalized 11-gonal (or hendecagonal) numbers: m*(9*m - 7)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...
48

%I #50 Sep 08 2022 08:45:59

%S 0,1,8,11,25,30,51,58,86,95,130,141,183,196,245,260,316,333,396,415,

%T 485,506,583,606,690,715,806,833,931,960,1065,1096,1208,1241,1360,

%U 1395,1521,1558,1691,1730,1870,1911,2058,2101,2255,2300,2461,2508,2676

%N Generalized 11-gonal (or hendecagonal) numbers: m*(9*m - 7)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...

%H Vincenzo Librandi, <a href="/A195160/b195160.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_, Sep 14 2011: (Start)

%F G.f.: x*(1+7*x+x^2)/((1+x)^2*(1-x)^3).

%F a(n) = (18*n*(n+1)+5*(2*n+1)*(-1)^n-5)/16.

%F a(2n) = A062728(n), a(2n-1) = A051682(n). (End)

%F Sum_{n>=1} 1/a(n) = 18/49 + 2*Pi*cot(2*Pi/9)/7. - _Vaclav Kotesovec_, Oct 05 2016

%t CoefficientList[Series[x (1 + 7 x + x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 60}], x] (* _Vincenzo Librandi_, Apr 09 2013 *)

%o (Magma) I:=[0, 1, 8, 11, 25]; [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..50]]; // _Vincenzo Librandi_, Apr 09 2013

%o (PARI) a(n)=(18*n*(n+1)+5*(2*n+1)*(-1)^n-5)/16 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Partial sums of A195159.

%Y Column 7 of A195152.

%Y Cf. A316672.

%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), this sequence (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), A303812 (k=28), A303815 (k=29), A316729 (k=30).

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Sep 10 2011