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A317300
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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.
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3
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0, 1, -3, 0, -8, -3, -15, -8, -24, -15, -35, -24, -48, -35, -63, -48, -80, -63, -99, -80, -120, -99, -143, -120, -168, -143, -195, -168, -224, -195, -255, -224, -288, -255, -323, -288, -360, -323, -399, -360, -440, -399, -483, -440, -528, -483, -575, -528, -624, -575, -675, -624, -728, -675, -783
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OFFSET
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0,3
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COMMENTS
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Taking the same formula with k = 1 we have A317301.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
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).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).
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LINKS
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FORMULA
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G.f.: x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = -n*(n + 4) / 4 for n even.
a(n) = -(n - 3)*(n + 1) / 4 for n odd.
(End)
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PROG
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(PARI) concat(0, Vec(x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ Colin Barker, Aug 01 2018
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CROSSREFS
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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).
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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