A266677 - OEIS

%I #24 Feb 02 2016 12:24:58

%S 0,-1,6,-16,34,-61,100,-152,220,-305,410,-536,686,-861,1064,-1296,

%T 1560,-1857,2190,-2560,2970,-3421,3916,-4456,5044,-5681,6370,-7112,

%U 7910,-8765,9680,-10656,11696,-12801,13974,-15216,16530,-17917,19380,-20920,22540

%N Alternating sum of hexagonal pyramidal numbers.

%C More generally, the ordinary generating function for the alternating sum of k-gonal pyramidal numbers is x*(1 + (3 - k)*x)/((x - 1)*(x + 1)^4).

%H Ilya Gutkovskiy, <a href="/A266677/a266677.pdf">Extended graphic representation</a>

%H OEIS Wiki, <a href="http://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalPyramidalNumber.html">Hexagonal Pyramidal Number</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-2,2,3,1).

%F G.f.: x*(1 - 3*x)/((x - 1)*(x + 1)^4).

%F a(n) = ((-1)^n*(2*n*(n + 2)*(4*n + 1) - 3) + 3)/24.

%F a(n) = Sum_{k = 0..n} (-1)^k*A002412(k).

%t Table[((-1)^n (2 n (n + 2) (4 n + 1) - 3) + 3)/24, {n, 0, 40}]

%t LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 6, -16, 34}, 40]

%o (PARI) concat(0, Vec(x*(1 - 3*x)/((x - 1)*(x + 1)^4) + O(x^50))) \\ _Michel Marcus_, Feb 02 2016

%Y Cf. A000292, A002412, A002717, A173196.

%K sign,easy

%O 0,3

%A _Ilya Gutkovskiy_, Feb 02 2016