A211807 -id:A211807 - cp's OEIS frontend

A211790 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k

1, 7, 1, 23, 7, 1, 54, 22, 7, 1, 105, 51, 22, 7, 1, 181, 97, 50, 22, 7, 1, 287, 166, 96, 50, 22, 7, 1, 428, 263, 163, 95, 50, 22, 7, 1, 609, 391, 255, 161, 95, 50, 22, 7, 1, 835, 554, 378, 253, 161, 95, 50, 22, 7, 1, 1111, 756, 534, 374, 252, 161, 95, 50, 22, 7

Offset: 1

Clark Kimberling, Apr 21 2012

...
Let R be the array in A211790 and let R' be the array in A211793.  Then R(k,n) + R'(k,n) = 3^(n-1).  Moreover, (row k of R) =(row k of A211796) for k>2, by Fermat's last theorem; likewise, (row k of R')=(row k of A211799) for k>2.
...
Generalizations:  Suppose that b,c,d are nonzero integers, and let U(k,n) be the number of ordered triples (w,x,y) with all terms in {1,...,n} and b*w*k  c*x^k+d*y^k, where the relation  is one of these: <, >=, <=, >.  What additional assumptions force the limiting row sequence to be essentially one of these: A002412, A000330, A016061, A174723, A051925?
In the following guide to related arrays and sequences, U(k,n) denotes the number of (w,x,y) as described in the preceding paragraph:
                                 first 3 rows          limiting row sequence
A211790:  w^k < x^k+y^k    A004068, A211635, A211650       A002412
A211793:  w^k >= x^k+y^k   A000292, A211636, A211651       A000330
A211796:  w^k <= x^k+y^k   A002413, A211634, A211650       A002412
A211799:  w^k > x^k+y^k    A000292, A211637, A211651       A000330
A211802:  2w^k < x^k+y^k   A182260, A211800, A211801       A016061
A211805:  2w^k >= x^k+y^k  A055232, A211803, A211804       A000330
A211808:  2w^k <= x^k+y^k  A055232, A211806, A211807       A174723
A182259:  2w^k > x^k+y^k   A182260, A211810, A211811       A051925

			Northwest corner:
  1, 7, 23, 54, 105, 181, 287, 428, 609
  1, 7, 22, 51,  97, 166, 263, 391, 554
  1, 7, 22, 50,  96, 163, 255, 378, 534
  1, 7, 22, 50,  95, 161, 253, 374, 528
  1, 7, 22, 50,  95, 161, 252, 373, 527
For n=2 and k>=1, the 7 triples (w,x,y) are (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2).

Cf. A004068 (row1), A211635 (row2), A211650 (row3), A002412.

Cf. A211793, A211796, A211799, A211802, A211805, A211808, A182259

z = 48;
t[k_, n_] := Module[{s = 0},
   (Do[If[w^k < x^k + y^k, s = s + 1],
       {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
Table[t[1, n], {n, 1, z}]  (* A004068 *)
Table[t[2, n], {n, 1, z}]  (* A211635 *)
Table[t[3, n], {n, 1, z}]  (* A211650 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A211790 *)
Table[n (n + 1) (4 n - 1)/6,
  {n, 1, z}] (* row-limit sequence, A002412 *)
(* Peter J. C. Moses, Apr 13 2012 *)

R(k,n) = n(n-1)(4n+1)/6 for 1<=k<=n, and

R(k,n) = Sum{Sum{floor[(x^k+y^k)^(1/k)] : 1<=x<=n, 1<=y<=n}} for 1<=k<=n.

A211808 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^k<=x^k+y

Original entry on oeis.org

1, 5, 1, 16, 5, 1, 36, 16, 5, 1, 69, 36, 16, 5, 1, 117, 69, 38, 16, 5, 1, 184, 119, 73, 38, 16, 5, 1, 272, 190, 123, 75, 38, 16, 5, 1, 385, 282, 194, 131, 75, 38, 16, 5, 1, 525, 399, 290, 204, 131, 75, 38, 16, 5, 1, 696, 547, 415, 300, 210, 131, 75, 38, 16, 5, 1

Offset: 1

Clark Kimberling, Apr 22 2012

Row 1:  A055232
Row 2:  A211806
Row 3:  A211807
Limiting row sequence: A000330
Let R be the array in A211808 and let R' be the array in A182259.  Then R(k,n)+R'(k,n)=3^(n-1).
See the Comments at A211790.

			Northwest corner:
1...5...16...36...69...117...184
1...5...16...36...69...119...190
1...5...16...38...73...123...194
1...5...16...38...75...131...204
1...5...16...38...75...131...210

Cf. A211790.

z = 48;
t[k_, n_] := Module[{s = 0},
   (Do[If[2 w^k <= x^k + y^k, s = s + 1],
       {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
Table[t[1, n], {n, 1, z}]  (* A055232 *)
Table[t[2, n], {n, 1, z}]  (* A211806 *)
Table[t[3, n], {n, 1, z}]  (* A211807 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1],
     {n, 1, 12}, {k, 1, n}]] (* A211808 *)
Table[k (4 k^2 - 3 k + 5)/6,
     {k, 1, z}] (* row-limit sequence, A174723 *)
(* Peter J. C. Moses, Apr 13 2012 *)

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A211790 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k

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