A211790 -id:A211790 - cp's OEIS frontend

A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.

1, 9, 17, 25, 41, 49, 57, 65, 81, 105, 113, 121, 137, 145, 153, 161, 193, 201, 225, 233, 249, 257, 265, 273, 289, 329, 337, 361, 377, 385, 393, 401, 433, 441, 449, 457, 505, 513, 521, 529, 545, 553, 561, 569, 585, 609, 617, 625, 657, 713, 753, 761

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

Suppose that S={-n,...,0,...,n} and that f(w,x,y,n) is a function, where w,x,y are in S. The number of ordered triples (w,x,y) satisfying f(w,x,y,n)=0, regarded as a function of n, is a sequence t of nonnegative integers. Sequences such as t/4 may also be integer sequences for all except certain initial values of n. In the following guide, such sequences are indicated in the related sequences column and may be included in the corresponding Mathematica programs.
...
sequence... f(w,x,y,n) ..... related sequences
A211415 ... w^2+x*y-1 ...... t+2, t/4, (t/4-1)/4
A211422 ... w^2+x*y ........ (t-1)/8, A120486
A211423 ... w^2+2x*y ....... (t-1)/4
A211424 ... w^2+3x*y ....... (t-1)/4
A211425 ... w^2+4x*y ....... (t-1)/4
A211426 ... 2w^2+x*y ....... (t-1)/4
A211427 ... 3w^2+x*y ....... (t-1)/4
A211428 ... 2w^2+3x*y ...... (t-1)/4
A211429 ... w^3+x*y ........ (t-1)/4
A211430 ... w^2+x+y ........ (t-1)/2
A211431 ... w^3+(x+y)^2 .... (t-1)/2
A211432 ... w^2-x^2-y^2 .... (t-1)/8
A003215 ... w+x+y .......... (t-1)/2, A045943
A202253 ... w+2x+3y ........ (t-1)/2, A143978
A211433 ... w+2x+4y ........ (t-1)/2
A211434 ... w+2x+5y ........ (t-1)/4
A211435 ... w+4x+5y ........ (t-1)/2
A211436 ... 2w+3x+4y ....... (t-1)/2
A211435 ... 2w+3x+5y ....... (t-1)/2
A211438 ... w+2x+2y ....... (t-1)/2, A118277
A001844 ... w+x+2y ......... (t-1)/4, A000217
A211439 ... w+3x+3y ........ (t-1)/2
A211440 ... 2w+3x+3y ....... (t-1)/2
A028896 ... w+x+y-1 ........ t/6, A000217
A211441 ... w+x+y-2 ........ t/3, A028387
A182074 ... w^2+x*y-n ...... t/4, A028387
A000384 ... w+x+y-n
A000217 ... w+x+y-2n
A211437 ... w*x*y-n ........ t/4, A007425
A211480 ... w+2x+3y-1
A211481 ... w+2x+3y-n
A211482 ... w*x+w*y+x*y-w*x*y
A211483 ... (n+w)^2-x-y
A182112 ... (n+w)^2-x-y-w
...
For the following sequences, S={1,...,n}, rather than
{-n,...,0,...n}. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.
A132188 ... w^2-x*y
A211506 ... w^2-x*y-n
A211507 ... w^2-x*y+n
A211508 ... w^2+x*y-n
A211509 ... w^2+x*y-2n
A211510 ... w^2-x*y+2n
A211511 ... w^2-2x*y ....... t/2
A211512 ... w^2-3x*y ....... t/2
A211513 ... 2w^2-x*y ....... t/2
A211514 ... 3w^2-x*y ....... t/2
A211515 ... w^3-x*y
A211516 ... w^2-x-y
A211517 ... w^3-(x+y)^2
A063468 ... w^2-x^2-y^2 .... t/2
A000217 ... w+x-y
A001399 ... w-2x-3y
A211519 ... w-2x+3y
A008810 ... w+2x-3y
A001399 ... w-2x-3y
A008642 ... w-2x-4y
A211520 ... w-2x+4y
A211521 ... w+2x-4y
A000115 ... w-2x-5y
A211522 ... w-2x+5y
A211523 ... w+2x-5y
A211524 ... w-3x-5y
A211533 ... w-3x+5y
A211523 ... w+3x-5y
A211535 ... w-4x-5y
A211536 ... w-4x+5y
A008812 ... w+4x-5y
A055998 ... w+x+y-2n
A074148 ... 2w+x+y-2n
A211538 ... 2w+2x+y-2n
A211539 ... 2w+2x-y-2n
A211540 ... 2w-3x-4y
A211541 ... 2w-3x+4y
A211542 ... 2w+3x-4y
A211543 ... 2w-3x-5y
A211544 ... 2w-3x+5y
A008812 ... 2w+3x-5y
A008805 ... w-2x-2y (repeated triangular numbers)
A001318 ... w-2x+2y
A000982 ... w+x-2y
A211534 ... w-3x-3y
A211546 ... w-3x+3y (triply repeated triangular numbers)
A211547 ... 2w-3x-3y (triply repeated squares)
A082667 ... 2w-3x+3y
A055998 ... w-x-y+2
A001399 ... w-2x-3y+1
A108579 ... w-2x-3y+n
...
Next, S={-n,...-1,1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated inequality. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.
A211545 ... w+x+y>0; recurrence degree: 4
A211612 ... w+x+y>=0
A211613 ... w+x+y>1
A211614 ... w+x+y>2
A211615 ... |w+x+y|<=1
A211616 ... |w+x+y|<=2
A211617 ... 2w+x+y>0; recurrence degree: 5
A211618 ... 2w+x+y>1
A211619 ... 2w+x+y>2
A211620 ... |2w+x+y|<=1
A211621 ... w+2x+3y>0
A211622 ... w+2x+3y>1
A211623 ... |w+2x+3y|<=1
A211624 ... w+2x+2y>0; recurrence degree: 6
A211625 ... w+3x+3y>0; recurrence degree: 8
A211626 ... w+4x+4y>0; recurrence degree: 10
A211627 ... w+5x+5y>0; recurrence degree: 12
A211628 ... 3w+x+y>0; recurrence degree: 6
A211629 ... 4w+x+y>0; recurrence degree: 7
A211630 ... 5w+x+y>0; recurrence degree: 8
A211631 ... w^2>x^2+y^2; all terms divisible by 8
A211632 ... 2w^2>x^2+y^2; all terms divisible by 8
A211633 ... w^2>2x^2+2y^2; all terms divisible by 8
...
Next, S={1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated relation.
A211634 ... w^2<=x^2+y^2
A211635 ... w^2A211790
A211636 ... w^2>=x^2+y^2
A211637 ... w^2>x^2+y^2
A211638 ... w^2+x^2+y^2A211639 ... w^2+x^2+y^2<=n
A211640 ... w^2+x^2+y^2>n
A211641 ... w^2+x^2+y^2>=n
A211642 ... w^2+x^2+y^2<2n
A211643 ... w^2+x^2+y^2<=2n
A211644 ... w^2+x^2+y^2>2n
A211645 ... w^2+x^2+y^2>=2n
A211646 ... w^2+x^2+y^2<3n
A211647 ... w^2+x^2+y^2<=3n
A063691 ... w^2+x^2+y^2=n
A211649 ... w^2+x^2+y^2=2n
A211648 ... w^2+x^2+y^2=3n
A211650 ... w^3A211790
A211651 ... w^3>x^3+y^3; see Comments at A211790
A211652 ... w^4A211790
A211653 ... w^4>x^4+y^4; see Comments at A211790

			a(1) counts these 9 triples: (-1,-1,1), (-1, 1,-1), (0, -1, 0), (0, 0, -1), (0,0,0), (0,0,1), (0,1,0), (1,-1,1), (1,1,-1).

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A120486.

t[n_] := t[n] = Flatten[Table[w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}] (* A211422 *)
(t - 1)/8                   (* A120486 *)

A182260 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w

Original entry on oeis.org

0, 3, 11, 28, 56, 99, 159, 240, 344, 475, 635, 828, 1056, 1323, 1631, 1984, 2384, 2835, 3339, 3900, 4520, 5203, 5951, 6768, 7656, 8619, 9659, 10780, 11984, 13275, 14655, 16128, 17696, 19363, 21131, 23004, 24984, 27075, 29279, 31600, 34040

Offset: 1

Clark Kimberling, Apr 22 2012

Also the number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w>x+y.
A182260(n)+A055232(n)=3^(n-1).
A182260 is row 1 of A211802 and also row 1 of A182259; see A211790 for a discussion and guide to related sequences.

			For n=2, the 3 triples (w,x,y) for which 2w<x+y are (1,1,2), (1,2,1), (1,2,2).  The 3 triples for which 2w>x+y are (2,1,1), (2,1,2), (2,2,1).

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A211790, A211802.

(See the program at A211802.)
LinearRecurrence[{3,-2,-2,3,-1},{0,3,11,28,56},50] (* Harvey P. Dale, Aug 10 2019 *)

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5).
From Colin Barker, May 06 2012: (Start)
a(n) = (-1+(-1)^n-2*n^2+4*n^3)/8.
G.f.: x^2*(3 + 2*x + x^2)/((1 - x)^4*(1 + x)). (End)

A182259 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

0, 3, 0, 11, 3, 0, 28, 11, 3, 0, 56, 28, 11, 3, 0, 99, 56, 26, 11, 3, 0, 159, 97, 52, 26, 11, 3, 0, 240, 153, 93, 50, 26, 11, 3, 0, 344, 230, 149, 85, 50, 26, 11, 3, 0, 475, 330, 222, 139, 85, 50, 26, 11, 3, 0, 635, 453, 314, 212, 133, 85, 50, 26, 11, 3, 0, 828

Offset: 1

Clark Kimberling, Apr 22 2012

Row 1:  A182260
Row 2:  A211810
Row 3:  A211811
Limiting row sequence: A051925
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 (with antidiagonals read from northeast to southwest):
0...3...11...28...56...99...159
0...3...11...28...56...97...153
0...3...11...26...52...93...149
0...3...11...26...50...85...139
0...3...11...26...50...85...133

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}]  (* A182260 *)
Table[t[2, n], {n, 1, z}]  (* A211810 *)
Table[t[3, n], {n, 1, z}]  (* A211811 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1],
    {n, 1, 12}, {k, 1, n}]] (* A182259 *)
Table[k (k - 1) (2 k + 5)/6,
    {k, 1, z}] (* row-limit sequence, A051925 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A211802 R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and 2*w^k < x^k + y^k; square array read by descending antidiagonals.

Original entry on oeis.org

0, 3, 0, 11, 3, 0, 28, 13, 3, 0, 56, 32, 13, 3, 0, 99, 64, 34, 13, 3, 0, 159, 113, 68, 34, 13, 3, 0, 240, 181, 117, 70, 34, 13, 3, 0, 344, 272, 187, 125, 70, 34, 13, 3, 0, 475, 388, 282, 197, 125, 70, 34, 13, 3, 0, 635, 535, 406, 292, 203, 125, 70, 34, 13, 3, 0

Offset: 1

Clark Kimberling, Apr 22 2012

Row 1:  A182260.
Row 2:  A211800.
Row 3:  A211801.
Limiting row sequence: A016061.
Let R be the array in this sequence and let R' be the array in A211805.  Then R(k,n) + R'(k,n) = 3^(n-1).
See the Comments at A211790.

			Northwest corner:
  0   3  11  28  56  99 159 240
  0   3  13  32  64 113 181 272
  0   3  13  34  68 117 187 282
  0   3  13  34  70 125 197 292
  0   3  13  34  70 125 203 302

Cf. A016061, A182260, A211790, A211800, A211801, A211802, A211805.

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}]  (* A182260 *)
Table[t[2, n], {n, 1, z}]  (* A211800 *)
Table[t[3, n], {n, 1, z}]  (* A211801 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1], {n, 1, 12},
                {k, 1, n}]] (* this sequence *)
Table[k (k - 1) (4 k + 1)/6, {k, 1,
  z}] (* row-limit sequence, A016061 *)
(* Peter J. C. Moses, Apr 13 2012 *)

Definition corrected by Georg Fischer, Sep 10 2022

A211805 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, 14, 5, 1, 69, 32, 14, 5, 1, 117, 61, 30, 14, 5, 1, 184, 103, 57, 30, 14, 5, 1, 272, 162, 99, 55, 30, 14, 5, 1, 385, 240, 156, 91, 55, 30, 14, 5, 1, 525, 341, 230, 146, 91, 55, 30, 14, 5, 1, 696, 465, 323, 220, 140, 91, 55, 30, 14, 5, 1, 900

Offset: 1

Clark Kimberling, Apr 22 2012

Row 1:  A055232
Row 2:  A211803
Row 3:  A211804
Limiting row sequence: A000330
Let R be the array in A211802 and let R' be the array in A211805.  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...14...32...61...103...162
1...5...14...30...57...99....156
1...5...14...30...55...91....146
1...5...14...30...55...91....140

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}]  (* A211803 *)
Table[t[3, n], {n, 1, z}]  (* A211804 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1], {n, 1, 12},
               {k, 1, n}]] (* A211805 *)
Table[k (k + 1) (2 k + 1)/6,
    {k, 1, z}] (* row-limit sequence, A000330 *)
(* Peter J. C. Moses, Apr 13 2012 *)

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 *)

A211796 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k<=x^k+y^k.

Original entry on oeis.org

1, 8, 1, 26, 7, 1, 60, 22, 7, 1, 115, 51, 22, 7, 1, 196, 99, 50, 22, 7, 1, 308, 168, 96, 50, 22, 7, 1, 456, 265, 163, 95, 50, 22, 7, 1, 645, 393, 255, 161, 95, 50, 22, 7, 1, 880, 556, 378, 253, 161, 95, 50, 22, 7, 1, 1166, 760, 534, 374, 252, 161, 95, 50, 22, 7

Offset: 1

Clark Kimberling, Apr 21 2012

Row 1:  A002413
Row 2:  A211634
Row 3:  A211650
Limiting row sequence: A002412
Let R be the array in A211796 and let R' be the array in A211799.  Then R(k,n)+R'(k,n)=3^(n-1).
See the Comments at A211790.

			Northwest corner:
1...8...26...60...115...196...308
1...7...22...51...99....168...265
1...7...22...50...96....163...255
1...7...22...50...95....161...253
1...7...22...50...95....161...252

Cf. A211790.

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}]  (* A002413 *)
Table[t[2, n], {n, 1, z}]  (* A211634 *)
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}]] (* A211796 *)
Table[k (k - 1) (2 k - 1)/6, {k, 1,
  z}] (* row-limit sequence, A002412 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A211799 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k<=x^k+y

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 4, 5, 1, 0, 10, 13, 5, 1, 0, 20, 26, 14, 5, 1, 0, 35, 48, 29, 14, 5, 1, 0, 56, 78, 53, 30, 14, 5, 1, 0, 84, 119, 88, 55, 30, 14, 5, 1, 0, 120, 173, 134, 90, 55, 30, 14, 5, 1, 0, 165, 240, 195, 138, 91, 55, 30, 14, 5, 1, 0, 220, 323, 270, 201, 139, 91

Offset: 1

Clark Kimberling, Apr 21 2012

Row 1:  A002292
Row 2:  A211637
Row 3:  A211651
Limiting row sequence: A000330
Let R be the array in A211796 and let R' be the array in A211799.  Then R(k,n)+R'(k,n)=3^(n-1).
See the Comments at A211790.

			Northwest corner:
0...0...1...4....10...20...35...56
0...1...5...13...26...48...78...119
0...1...5...14...29...53...88...134
0...1...5...14...30...55...90...138
0...1...5...14...30...55...91...139

Cf. A211790.

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}]  (* A000292 *)
Table[t[2, n], {n, 1, z}]  (* A211637 *)
Table[t[3, n], {n, 1, z}]  (* A211651 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1],
    {n, 1, 12}, {k, 1, n}]] (* A211799 *)
Table[k (k - 1) (2 k - 1)/6,
    {k, 1, z}] (* row-limit sequence, A000330 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A211801 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^3

Original entry on oeis.org

0, 3, 13, 34, 68, 117, 187, 282, 406, 559, 743, 966, 1232, 1545, 1899, 2304, 2764, 3285, 3869, 4512, 5222, 6005, 6867, 7812, 8828, 9931, 11125, 12412, 13798, 15271, 16847, 18532, 20330, 22239, 24255, 26394, 28660, 31055, 33573, 36222

Offset: 1

Clark Kimberling, Apr 22 2012

nonn

Row 3 of A211802; see A211790 for a discussion and guide to related sequences.

Cf. A211790, A211802.

Mathematica
```
(See the program at A211802.)
```

A211807 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^3<=x^3+y^3.

Original entry on oeis.org

1, 5, 16, 38, 73, 123, 194, 290, 415, 569, 754, 978, 1245, 1559, 1914, 2320, 2781, 3303, 3888, 4532, 5243, 6027, 6890, 7836, 8853, 9957, 11152, 12440, 13827, 15301, 16878, 18564, 20363, 22273, 24290, 26430, 28697, 31093, 33612, 36262

Offset: 1

Clark Kimberling, Apr 22 2012

nonn

Row 3 of A211808; see A211790 for a discussion and guide to related sequences.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A211790, A211808.

f:= proc(n) local x;
  n + add(2*floor(((x^3+n^3)/2)^(1/3)), x=1..n-1)
end proc:
ListTools:-PartialSums(map(f,[$1..50])); # Robert Israel, Jan 26 2025

Mathematica
```
(See the program at A211808.)
```

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A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.

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A182260 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w

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A182259 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^k<=x^k+y

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A211802 R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and 2*w^k < x^k + y^k; square array read by descending antidiagonals.

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A211805 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^k>=x^k+y

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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

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A211796 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k<=x^k+y^k.

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A211799 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k<=x^k+y

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A211801 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^3

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