A031876 - OEIS

%I #27 Aug 23 2017 10:58:59

%S 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,

%T 45,48,51,54,57,60,63,66,69,72,75,78,81,84,87,90,93,96,99,102,105,108,

%U 111,114,117,120,123,126,129,132,135,138,141,144,147,150,153,156,160

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

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

%H G. C. Greubel, <a href="/A031876/b031876.txt">Table of n, a(n) for n = 0..1000</a>

%F 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.

%F 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

%F G.f.: Sum_{k>=1} x^(k^3)/(1 - x)^2. - _Ilya Gutkovskiy_, Dec 22 2016

%F a(n) = (3/4)*n^(4/3) + O(n). - _Charles R Greathouse IV_, Aug 23 2017

%t Accumulate[Floor[Surd[Range[0,70],3]]] (* _Harvey P. Dale_, Nov 03 2013 *)

%t Table[Sum[Floor[i^(1/3)], {i,0,n}], {n,0,50}] (* _G. C. Greubel_, Dec 22 2016 *)

%o (PARI) a(n) = sum(k=1, n, sqrtnint(k, 3)); \\ _Michel Marcus_, Mar 12 2016

%o (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

%Y Cf. A022554.

%K nonn,easy

%O 0,3

%A Michel Tixier (tixier(AT)dyadel.net)