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A195313 Generalized 13-gonal numbers: m*(11*m-9)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...

%I A195313 #65 Sep 08 2022 08:45:59
%S A195313 0,1,10,13,31,36,63,70,106,115,160,171,225,238,301,316,388,405,486,
%T A195313 505,595,616,715,738,846,871,988,1015,1141,1170,1305,1336,1480,1513,
%U A195313 1666,1701,1863,1900,2071,2110,2290,2331,2520,2563,2761,2806,3013,3060,3276
%N A195313 Generalized 13-gonal numbers: m*(11*m-9)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...
%C A195313 Also generalized tridecagonal numbers or generalized triskaidecagonal numbers.
%C A195313 Also A211013 and positive terms of A051865 interleaved. - _Omar E. Pol_, Aug 04 2012
%C A195313 Numbers k for which 88*k + 81 is a square. - _Bruno Berselli_, Jul 10 2018
%H A195313 Vincenzo Librandi, <a href="/A195313/b195313.txt">Table of n, a(n) for n = 0..10000</a>
%H A195313 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A195313 From _Bruno Berselli_, Sep 15 2011: (Start)
%F A195313 G.f.: x*(1 + 9*x + x^2)/((1 + x)^2*(1 - x)^3).
%F A195313 a(n) = (22*n*(n + 1) + 7*(2*n + 1)*(-1)^n - 7)/16.
%F A195313 a(n) - a(n-2) = A175885(n). (End)
%F A195313 Sum_{n>=1} 1/a(n) = 22/81 + 2*Pi*cot(2*Pi/11)/9. - _Vaclav Kotesovec_, Oct 05 2016
%p A195313 a:= n-> (m-> m*(11*m-9)/2)(-ceil(n/2)*(-1)^n):
%p A195313 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jul 10 2018
%t A195313 lim = 50; Sort[Table[n*(11*n - 9)/2, {n, -lim, lim}]] (* _T. D. Noe_, Sep 15 2011 *)
%t A195313 Accumulate[With[{nn=30},Riffle[9Range[0,nn],Range[1,2nn+1,2]]]] (* _Harvey P. Dale_, Sep 24 2011 *)
%o A195313 (Magma) [(22*n*(n+1)+7*(2*n+1)*(-1)^n-7)/16: n in [0..50]]; // _Vincenzo Librandi_, Sep 16 2011
%o A195313 (Magma) A195313:=func<n | n*(11*n-9)/2>; [0] cat [A195313(n*m): m in [1,-1], n in [1..25]]; // _Bruno Berselli_, Nov 13 2012
%o A195313 (PARI) a(n)=(22*n*(n+1)+7*(2*n+1)*(-1)^n-7)/16 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A195313 Partial sums of A195312.
%Y A195313 Column 9 of A195152.
%Y A195313 Cf. A316672.
%Y A195313 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), this sequence (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 A195313 nonn,easy
%O A195313 0,3
%A A195313 _Omar E. Pol_, Sep 14 2011