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%I #32 Aug 01 2018 10:56:52
%S 0,1,-3,0,-8,-3,-15,-8,-24,-15,-35,-24,-48,-35,-63,-48,-80,-63,-99,
%T -80,-120,-99,-143,-120,-168,-143,-195,-168,-224,-195,-255,-224,-288,
%U -255,-323,-288,-360,-323,-399,-360,-440,-399,-483,-440,-528,-483,-575,-528,-624,-575,-675,-624,-728,-675,-783
%N Sequence obtained by taking the general formula for generalized k-gonal numbers: m*((k - 2)*m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.
%C Taking the same formula with k = 1 we have A317301.
%C Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
%C Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
%C Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
%C Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).
%H Colin Barker, <a href="/A317300/b317300.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(n) = -A174474(n+1).
%F From _Colin Barker_, Aug 01 2018: (Start)
%F G.f.: x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2).
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
%F a(n) = -n*(n + 4) / 4 for n even.
%F a(n) = -(n - 3)*(n + 1) / 4 for n odd.
%F (End)
%o (PARI) concat(0, Vec(x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ _Colin Barker_, Aug 01 2018
%Y Row 0 of A303301.
%Y Cf. A001057, A008795, A008794, A174474, A317301.
%Y 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), A195313 (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 sign,easy
%O 0,3
%A _Omar E. Pol_, Jul 29 2018