A031876 a(n) = Sum_{k=0..n} floor(k^(1/3)).

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 160

Offset: 0

Author

Michel Tixier (tixier(AT)dyadel.net)

Keywords

• nonn
• easy

References

• K. H. Rosen, Discrete Mathematics and Its Application, 6th Edition, McGraw-Hill, 2007, Ex. 26 of section 2.4.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A022554.

Programs

Mathematica

Accumulate[Floor[Surd[Range[0,70],3]]] (* Harvey P. Dale, Nov 03 2013 *)
Table[Sum[Floor[i^(1/3)], {i,0,n}], {n,0,50}] (* G. C. Greubel, Dec 22 2016 *)

PARI

a(n) = sum(k=1, n, sqrtnint(k, 3)); \\ Michel Marcus, Mar 12 2016

PARI

a(n)=my(t=sqrtnint(n,3)); t*(4*n-t^3-2*t^2-t+4)/4 \\ Charles R Greathouse IV, Aug 23 2017

Formula

a(0) = 0, a(1) = 1, a(n) = 2*a(n-1) - a(n-2) if n not a perfect cube, else a(n) = 2*a(n-1) - a(n-2) + 1 if n is a perfect cube.

a(n) = -1/4*floor(n^(1/3))*(floor(n^(1/3))^3+2*floor(n^(1/3))^2+floor(n^(1/3))-4*(n+1)). - John M. Campbell, Mar 22 2016

G.f.: Sum_{k>=1} x^(k^3)/(1 - x)^2. - Ilya Gutkovskiy, Dec 22 2016

a(n) = (3/4)*n^(4/3) + O(n). - Charles R Greathouse IV, Aug 23 2017