A248575 Rounded sums of the non-integer cube roots of n, as partitioned by the integer roots: round[sum(j from n^3+1 to (n+1)^3-1, j^(1/3))].

0, 10, 46, 128, 272, 498, 822, 1264, 1840, 2570, 3470, 4560, 5856, 7378, 9142, 11168, 13472, 16074, 18990, 22240, 25840, 29810, 34166, 38928, 44112, 49738, 55822, 62384, 69440, 77010, 85110, 93760, 102976, 112778, 123182, 134208, 145872, 158194, 171190, 184880, 199280, 214410, 230286, 246928, 264352, 282578, 301622

Offset

0,2

Comments

The fractional portions of each sum converge to 1/4 and 3/4, alternately.

The corresponding sums for square roots are given by A014105.

See A247112 for additional references to similar sequences and a conjecture.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Formula

a(n) = round[sum(j from n^3+1 to (n+1)^3-1, j^(1/3))].

a(n) = ((n+1)*a(n-2) + 3*a(n-1) - 30)/(n-2) - 20.

(Thanks to Mathematica for finding the recursive formula from the first 12 terms, as a DifferenceRoot, reformatted here for OEIS format and verified to n = 100. I could not "coax" Mathematica to produce a simple non-recursive formula, but I suspect one exists.)

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5), for n > 5. This implies that the digital roots (A010888) of the terms are cyclic with a period of eighteen. - Ivan N. Ianakiev, Dec 13 2014

a(n) = (1-(-1)^n+8*n+18*n^2+12*n^3)/4. - Colin Barker, Dec 30 2014

G.f.: 2*x*(5*x^2+8*x+5) / ((x-1)^4*(x+1)). - Colin Barker, Dec 30 2014

Mathematica

Below are tables for both formulas, with the former showing the fractional portion:
Table[AccountingForm[N[Sum[j^(1/3), {j, n^3 + 1, (n + 1)^3 - 1}], 10]], {n, 0, 50}]
RecurrenceTable [{a[n] == ((n + 1)*a[n - 2]  + 3* a[n - 1] - 30)/(n - 2) - 20,
  a[1] == 10, a[2] == 46}, a, {n, 1, 50}]

Prog

(PARI) a(n) = round(sum(j=n^3+1, (n+1)^3-1, j^(1/3))); \\ Michel Marcus, Dec 09 2014
(PARI) concat(0, Vec(2*x*(5*x^2+8*x+5)/((x-1)^4*(x+1)) + O(x^100))) \\ Colin Barker, Dec 30 2014

Crossrefs

Cf. A247112, A014105.

Sequence in context: A007940 A082604 A281550 * A341404 A320697 A081583

Adjacent sequences: A248572 A248573 A248574 * A248576 A248577 A248578

Keyword

nonn,easy

Author

Richard R. Forberg, Dec 02 2014

Status

approved