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%I #25 Sep 08 2022 08:46:15
%S 0,-1,10,-20,38,-57,84,-112,148,-185,230,-276,330,-385,448,-512,584,
%T -657,738,-820,910,-1001,1100,-1200,1308,-1417,1534,-1652,1778,-1905,
%U 2040,-2176,2320,-2465,2618,-2772,2934,-3097,3268,-3440,3620,-3801,3990,-4180
%N Alternating sum of 11-gonal (or hendecagonal) numbers.
%H G. C. Greubel, <a href="/A266087/b266087.txt">Table of n, a(n) for n = 0..5000</a>
%H OEIS Wiki, <a href="http://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-2,0,2,1).
%F G.f.: -x*(1 - 8*x)/((1 - x)*(1 + x)^3).
%F a(n) = ((18*n^2 + 4*n - 7)*(-1)^n + 7)/8.
%F a(n) = Sum_{k = 0..n} (-1)^k*A051682(k).
%F Lim_{n -> infinity} a(n + 1)/a(n) = -1.
%F E.g.f.: (1/4)*(9*x^2 - 11*x)*cosh(x) - (1/4)*(9*x^2 - 11*x - 7)*sinh(x). - _G. C. Greubel_, Jan 27 2016
%t Table[((18 n^2 + 4 n - 7) (-1)^n + 7)/8, {n, 0, 43}]
%t CoefficientList[Series[(x - 8 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* _Vincenzo Librandi_, Dec 21 2015 *)
%t Accumulate[Times@@@Partition[Riffle[PolygonalNumber[11,Range[0,50]],{1,-1},{2,-1,2}],2]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{-2,0,2,1},{0,-1,10,-20},50] (* _Harvey P. Dale_, Aug 27 2019 *)
%o (Magma) [(18*(-1)^n*n^2 + 4*(-1)^n*n - 7*(-1)^n + 7)/8: n in [0..50]]; // _Vincenzo Librandi_, Dec 21 2015
%o (PARI) x='x+O('x^100); concat(0, Vec(-x*(1-8*x)/((1-x)*(1+x)^3))) \\ _Altug Alkan_, Dec 21 2015
%Y Cf. A006578, A007586, A035608, A051682, A083392, A089594.
%K sign,easy
%O 0,3
%A _Ilya Gutkovskiy_, Dec 21 2015
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