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%I #20 Sep 12 2022 11:43:59
%S 1,7,22,51,97,164,258,382,541,741,982,1271,1611,2008,2466,2986,3577,
%T 4241,4982,5807,6715,7714,8808,10000,11297,12701,14217,15848,17600,
%U 19477,21482,23620,25895,28313,30879,33592,36460,39487,42678,46036
%N a(n) = Sum_{y=1..n} Sum_{x=1..n} floor((x^k + y^k)^(1/k)) with k = 3.
%C Row 3 of A211798.
%H David A. Corneth, <a href="/A211792/b211792.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{y=1..n} Sum_{x=1..n} floor((x^3 + y^3)^(1/3)).
%F a(n) = a(n-1) + floor((2*n^3)^(1/3)) + 2*Sum_{i = 1..n-1} floor((n^3 + i^3)^(1/3)) for n >= 2 and a(1) = 1. - _David A. Corneth_, Sep 12 2022
%e For a(3) we get the floor() values (1+2+3) + (2+2+3) + (3+3+3) = 22.
%t f[x_, y_, k_] := f[x, y, k] = Floor[(x^k + y^k)^(1/k)]
%t t[k_, n_] := Sum[Sum[f[x, y, k], {x, 1, n}], {y, 1, n}]
%t Table[t[1, n], {n, 1, 45}] (* 2*A002411 *)
%t Table[t[2, n], {n, 1, 45}] (* A211791 *)
%t Table[t[3, n], {n, 1, 45}] (* A211792 *)
%t TableForm[Table[t[k, n], {k, 1, 12},
%t {n, 1, 16}]] (* A211798 *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]]
%o (PARI) first(n) = { res = vector(n); res[1] = 1; for(i = 2, n, i3 = i^3; s = sum(j = 1, i-1, sqrtnint(i3 + j^3, 3)); res[i] = res[i-1] + sqrtnint(2*i3, 3) + 2*s; ); res } \\ _David A. Corneth_, Sep 12 2022
%Y Cf. A211791, A211798.
%K nonn
%O 1,2
%A _Clark Kimberling_, Apr 26 2012
%E Definition changed by _Georg Fischer_, Sep 10 2022
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