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A248621
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Floor of sums of the squares of the non-integer cube roots of n, as partitioned by the integer roots: floor[sum(j from n^3+1 to (n+1)^3-1, j^(2/3))].
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2
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0, 16, 120, 456, 1240, 2760, 5376, 9520, 15696, 24480, 36520, 52536, 73320, 99736, 132720, 173280, 222496, 281520, 351576, 433960, 530040, 641256, 769120, 915216, 1081200, 1268800, 1479816, 1716120, 1979656, 2272440, 2596560, 2954176, 3347520, 3778896, 4250680
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OFFSET
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0,2
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COMMENTS
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The fractional portion of each sum converges to 1/10.
See A248575 for the corresponding sums of the cube root.
See A247112 for the cube of the square roots, other references and a conjecture.
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LINKS
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Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
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FORMULA
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a(n) = floor[sum(j from n^3+1 to (n+1)^3-1, j^(2/3))].
a(n) = 2*n + 5*n^2 + 6*n^3 + 3*n^4.
G.f.: -8*x*(x+2)*(2*x+1) / (x-1)^5. - Colin Barker, Dec 30 2014
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MATHEMATICA
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Table[AccountingForm[N[Sum[j^(2/3), {j, n^3 + 1, (n + 1)^3 - 1}], 50]], {n, 0, 50}]
Table[2 n + 5 n^2 + 6 n^3 + 3 n^4, {n, 0, 50}]
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PROG
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(PARI) a(n) = floor(sum(j=n^3+1, (n+1)^3-1, j^(2/3))); \\ Michel Marcus, Dec 22 2014
(PARI) concat(0, Vec(-8*x*(x+2)*(2*x+1)/(x-1)^5 + O(x^100))) \\ Colin Barker, Dec 30 2014
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CROSSREFS
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Cf. A248575, A247112.
Sequence in context: A304469 A316286 A306050 * A190049 A317227 A138571
Adjacent sequences: A248618 A248619 A248620 * A248622 A248623 A248624
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KEYWORD
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nonn,easy
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AUTHOR
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Richard R. Forberg, Dec 02 2014
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STATUS
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approved
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