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A266086 Alternating sum of 9-gonal (or nonagonal) numbers.

%I A266086 #17 Feb 16 2025 08:33:28
%S A266086 0,-1,8,-16,30,-45,66,-88,116,-145,180,-216,258,-301,350,-400,456,
%T A266086 -513,576,-640,710,-781,858,-936,1020,-1105,1196,-1288,1386,-1485,
%U A266086 1590,-1696,1808,-1921,2040,-2160,2286,-2413,2546,-2680,2820,-2961,3108,-3256,3410
%N A266086 Alternating sum of 9-gonal (or nonagonal) numbers.
%H A266086 G. C. Greubel, <a href="/A266086/b266086.txt">Table of n, a(n) for n = 0..5000</a>
%H A266086 OEIS Wiki, <a href="http://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>
%H A266086 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NonagonalNumber.html">Nonagonal Number</a>
%H A266086 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-2,0,2,1).
%F A266086 G.f.: -x*(1 - 6*x)/((1 - x)*(1 + x)^3).
%F A266086 a(n) = ((14*n^2 + 4*n - 5)*(-1)^n + 5)/8.
%F A266086 a(n) = Sum_{k = 0..n} (-1)^k*A001106(k).
%F A266086 Lim_{n -> infinity} a(n + 1)/a(n) = -1.
%t A266086 Table[((14 n^2 + 4 n - 5) (-1)^n + 5)/8, {n, 0, 44}]
%t A266086 CoefficientList[Series[(x - 6 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* _Vincenzo Librandi_, Dec 21 2015 *)
%o A266086 (Magma) [(14*(-1)^n*n^2 + 4*(-1)^n*n - 5*(-1)^n + 5)/8: n in [0..50]]; // _Vincenzo Librandi_, Dec 21 2015
%o A266086 (PARI) x='x+O('x^100); concat(0, Vec(-x*(1-6*x)/((1-x)*(1+x)^3))) \\ _Altug Alkan_, Dec 21 2015
%Y A266086 Cf. A001106, A006578, A007584, A035608, A083392, A089594.
%K A266086 sign,easy
%O A266086 0,3
%A A266086 _Ilya Gutkovskiy_, Dec 21 2015