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A294604
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Number of ordinary double points of a family of threefolds.
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0
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10, 41, 120, 283, 566, 1029, 1738, 2745, 4150, 6049, 8504, 11661, 15646, 20525, 26496, 33715, 42246, 52345, 64198, 77861, 93654, 111793, 132320, 155625, 181954, 211329, 244216, 280891, 321350, 366141, 415570, 469601, 528870, 593713, 664056, 740629, 823798, 913445, 1010400, 1115059, 1227254
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OFFSET
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3,1
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COMMENTS
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The degree-n projective algebraic threefolds have been obtained from a class of polynomials introduced for the construction of nodal surfaces. The threefolds have ordinary double points as their only singularities.
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LINKS
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FORMULA
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a(n) = (1/18)*(7*n^4 - 24*n^3 + 39*n^2 - 36*n + 18) if n is divisible by 3; a(n) = (1/18)*(7*n^4 - 24*n^3 + 37*n^2 - 30*n + 10) otherwise. For n = 3, 4, 5, ...
G.f.: x^3*(10 + 21*x + 48*x^2 + 54*x^3 + 57*x^4 + 36*x^5 + 24*x^6 + x^7 + 2*x^8 - 2*x^9 + x^10) / ((1 - x)^5*(1 + x + x^2)^3).
a(n) = 2*a(n-1) - a(n-2) + 3*a(n-3) - 6*a(n-4) + 3*a(n-5) - 3*a(n-6) + 6*a(n-7) - 3*a(n-8) + a(n-9) - 2*a(n-10) + a(n-11) for n > 10.
(End)
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MAPLE
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alpha := n -> (7*n^4-24*n^3+39*n^2-36*n+18)/18:
a := n -> `if`(modp(n, 3)=0, alpha(n), alpha(n)-((n-2)^2+n)/9):
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MATHEMATICA
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alpha[n_] := (7*n^4 - 24*n^3 + 39*n^2 - 36*n + 18)/18;
a[n_] := If[Mod[n, 3] == 0, alpha[n], alpha[n] - ((n-2)^2 + n)/9];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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