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A245032
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a(n) = 27*(n - 6)^2 + 4*(n - 6)^3 = ((n - 6)^2)*(4*n + 3).
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1
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108, 175, 176, 135, 76, 23, 0, 31, 140, 351, 688, 1175, 1836, 2695, 3776, 5103, 6700, 8591, 10800, 13351, 16268, 19575, 23296, 27455, 32076, 37183, 42800, 48951, 55660, 62951, 70848, 79375, 88556, 98415, 108976, 120263, 132300, 145111, 158720, 173151, 188428
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OFFSET
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0,1
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COMMENTS
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The discriminant D of the Cardano Tartaglia cubic equation x^3 + p*x +q = 0 is D = -27*q^2 - 4*p^3, D < 0. Let D = (-1)*D then D = 27*q^2 + 4*p^3, D > 0, q = p > -6. So D = 27*(n - 6)^2 + 4*(n - 6)^3, D > 0, q = p = n, n >= 0 for all real solutions of D as positive integers. The case D < 0 is treated in A245033. Remark: ((n - 6)^2)*(4*n + 3) = ((6 - n)^2)*(4*n + 3).
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LINKS
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FORMULA
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a(n) = 27*(n - 6)^2 + 4*(n - 6)^3 = ((n - 6)^2)*(4*n + 3).
G.f.: (49*x^3+124*x^2-257*x+108) / (x-1)^4. - Colin Barker, Jul 11 2014
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EXAMPLE
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n = 0, a(0) = 27*(-6)^2 + 4*(-6)^3 = 27*36 + 4*(-216) = 108;
n = 7, a(7) = 27*(1)^2 + 4*(1)^3 = 27 + 4 = 31.
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MAPLE
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {108, 175, 176, 135}, 50] (* Harvey P. Dale, Oct 13 2019 *)
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PROG
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(PARI) vector(100, n, (n-7)^2*(4*n-1)) \\ Colin Barker, Jul 11 2014
(PARI) Vec((49*x^3+124*x^2-257*x+108)/(x-1)^4 + O(x^100)) \\ Colin Barker, Jul 11 2014
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CROSSREFS
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Cf. A028347 (Discriminant of quadratic equation : a(n) = n^2 - 4*n, n > 2), A245033 (a(n) = 4*(n + 7)^3 - 27*(n + 7)^2 = (4*n +1)*(n+7)^2), A245035 (a(n) = (-1)*( -27*(prime(n) - 7)^2 - 4*(prime(n) - 7)^3 ) = (prime(n) - 7)^2 * (4* prime(n) - 1) with n >= 1), A245036 (a(n) = 4*prime(n)^3 - 27*prime(n)^2 = (prime(n)^2)*[4*prime(n) - 27], n >= 4), A004767 (4*n+3).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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