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%I #34 Sep 08 2022 08:46:09
%S 0,8,67,267,746,1690,3333,5957,9892,15516,23255,33583,47022,64142,
%T 85561,111945,144008,182512,228267,282131,345010,417858,501677,597517,
%U 706476,829700,968383,1123767,1297142,1489846,1703265,1938833,2198032,2482392,2793491
%N Floor of sums of the cubes of the non-integer square roots of n, as partitioned by the integer roots: floor( sum( j from n^2+1 to (n+1)^2-1, j^(3/2) ) ).
%C The fractional portion of each sum converges to 1/40 as n --> infinity.
%C The corresponding sum for square roots of n is given by A014105 or n*(2n+1) with a fractional portion converging to 1/6.
%C See A248575 for the corresponding sums for the cube roots.
%C See A248621 for the corresponding sums of squares of the cube roots.
%C See A248698 for the corresponding sum of the fourth roots.
%C Conjecture: the corresponding sums for all fractional (rational) powers of n (e.g., 5/2, 7/2, 9/2, ..., 1/3, 2/3, 4/3, ..., 1/4, 3/4, 5/4, ..., 1/5, 2/5, 3/5, ...) will have polynomial integer formulas or recursive integer formulas for their floor, ceiling and/or rounded values, with convergence to a rational fractional portion, with possibly multiple fractional values in a repeating pattern as they converge. This was clear for some additional examples, less so for higher-order examples.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = floor( sum( j = n^2+1..(n+1)^2-1, j^(3/2) ) ).
%F a(n) = n *(4*n^3 + 6*n^2 + 5*n + 1)/2.
%F G.f.: x*(8 + 27*x + 12*x^2 + x^3)/(1 - x)^5. [_Bruno Berselli_, Dec 03 2014]
%t Table[1/2 (n + 5 n^2 + 6 n^3 + 4 n^4), {n, 0, 50}]
%t Table[N[Sum[j^(3/2), {j, n^2 + 1, (n + 1)^2 - 1}], 10], {n, 0, 50}]
%o (Magma) [n eq 0 select 0 else Floor(&+[j^(3/2): j in [n^2+1..(n+1)^2-1]]): n in [0..50]]; // _Bruno Berselli_, Dec 03 2014
%Y Cf. A014105, A248575, A248621, A248698.
%K nonn,easy
%O 0,2
%A _Richard R. Forberg_, Dec 02 2014
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