A266088 - OEIS

%I #26 Sep 08 2022 08:46:15

%S 0,-1,11,-22,42,-63,93,-124,164,-205,255,-306,366,-427,497,-568,648,

%T -729,819,-910,1010,-1111,1221,-1332,1452,-1573,1703,-1834,1974,-2115,

%U 2265,-2416,2576,-2737,2907,-3078,3258,-3439,3629,-3820,4020,-4221,4431,-4642

%N Alternating sum of 12-gonal (or dodecagonal) numbers.

%C More generally, the ordinary generating function for the alternating sum of k-gonal numbers is -x*(1 - (k - 3)*x)/((1 - x)*(1 + x)^3).

%H G. C. Greubel, <a href="/A266088/b266088.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 - 9*x)/((1 - x)*(1 + x)^3).

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

%F a(n) = Sum_{k = 0..n} (-1)^k*A051624(k).

%F Lim_{n -> infinity} a(n + 1)/a(n) = -1.

%t Table[1 + (-1)^n (5 n^2 + n - 2)/2, {n, 0, 43}]

%t CoefficientList[Series[-x (1 - 9 x)/((1 - x) (1 + x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Dec 21 2015 *)

%o (Magma) [1+(-1)^n*(5*n^2+n-2)/2: n in [0..50]]; // _Vincenzo Librandi_, Dec 21 2015

%o (PARI) x='x+O('x^100); concat(0, Vec(-x*(1-9*x)/((1-x)*(1+x)^3))) \\ _Altug Alkan_, Dec 21 2015

%Y Cf. A006578, A007587, A035608, A051624, A083392, A089594.

%K sign,easy

%O 0,3

%A _Ilya Gutkovskiy_, Dec 21 2015