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A248621 Floor of sums of the squares of the non-integer cube roots of n, as partitioned by the integer roots: floor(Sum_{j=n^3+1..(n+1)^3-1} j^(2/3)).

%I A248621 #41 Jul 14 2024 08:50:07
%S A248621 0,16,120,456,1240,2760,5376,9520,15696,24480,36520,52536,73320,99736,
%T A248621 132720,173280,222496,281520,351576,433960,530040,641256,769120,
%U A248621 915216,1081200,1268800,1479816,1716120,1979656,2272440,2596560,2954176,3347520,3778896,4250680
%N A248621 Floor of sums of the squares of the non-integer cube roots of n, as partitioned by the integer roots: floor(Sum_{j=n^3+1..(n+1)^3-1} j^(2/3)).
%C A248621 The fractional portion of each sum converges to 1/10.
%C A248621 See A248575 for the corresponding sums of the cube root.
%C A248621 See A247112 for the cube of the square roots, other references and a conjecture.
%H A248621 Colin Barker, <a href="/A248621/b248621.txt">Table of n, a(n) for n = 0..1000</a>
%H A248621 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A248621 a(n) = floor(Sum_{j=n^3+1..(n+1)^3-1} j^(2/3)).
%F A248621 a(n) = 2*n + 5*n^2 + 6*n^3 + 3*n^4.
%F A248621 G.f.: -8*x*(x+2)*(2*x+1) / (x-1)^5. - _Colin Barker_, Dec 30 2014
%F A248621 E.g.f.: exp(x)*x*(16 + 44*x + 24*x^2 + 3*x^3). - _Stefano Spezia_, Jul 13 2024
%t A248621 Table[2 n + 5 n^2 + 6 n^3 + 3 n^4, {n, 0, 50}]
%o A248621 (PARI) a(n) = floor(sum(j=n^3+1, (n+1)^3-1, j^(2/3))); \\ _Michel Marcus_, Dec 22 2014
%o A248621 (PARI) concat(0, Vec(-8*x*(x+2)*(2*x+1)/(x-1)^5 + O(x^100))) \\ _Colin Barker_, Dec 30 2014
%Y A248621 Cf. A247112, A248575, A248698.
%K A248621 nonn,easy
%O A248621 0,2
%A A248621 _Richard R. Forberg_, Dec 02 2014