A211792 a(n) = Sum_{y=1..n} Sum_{x=1..n} floor((x^k + y^k)^(1/k)) with k = 3.

1, 7, 22, 51, 97, 164, 258, 382, 541, 741, 982, 1271, 1611, 2008, 2466, 2986, 3577, 4241, 4982, 5807, 6715, 7714, 8808, 10000, 11297, 12701, 14217, 15848, 17600, 19477, 21482, 23620, 25895, 28313, 30879, 33592, 36460, 39487, 42678, 46036

Offset

1,2

Comments

Row 3 of A211798.

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{y=1..n} Sum_{x=1..n} floor((x^3 + y^3)^(1/3)).

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

Example

For a(3) we get the floor() values (1+2+3) + (2+2+3) + (3+3+3) = 22.

Mathematica

f[x_, y_, k_] := f[x, y, k] = Floor[(x^k + y^k)^(1/k)]
t[k_, n_] := Sum[Sum[f[x, y, k], {x, 1, n}], {y, 1, n}]
Table[t[1, n], {n, 1, 45}]  (* 2*A002411 *)
Table[t[2, n], {n, 1, 45}]  (* A211791 *)
Table[t[3, n], {n, 1, 45}]  (* A211792 *)
TableForm[Table[t[k, n], {k, 1, 12},
                 {n, 1, 16}]] (* A211798 *)
Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]]

Prog

(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

Crossrefs

Cf. A211791, A211798.

Sequence in context: A002412 A211652 A211650 * A211635 A211634 A184629

Adjacent sequences: A211789 A211790 A211791 * A211793 A211794 A211795

Keyword

nonn

Author

Clark Kimberling, Apr 26 2012

Extensions

Definition changed by Georg Fischer, Sep 10 2022

Status

approved