cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Apr 21 2012

Keywords

Comments

...
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

Examples

			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).

Crossrefs

Cf. A004068 (row1), A211635 (row2), A211650 (row3), A002412.
Cf. A211793, A211796, A211799, A211802, A211805, A211808, A182259

Programs

  • Mathematica
    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 *)

Formula

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.

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

Views

Author

Clark Kimberling, Apr 22 2012

Keywords

Comments

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.

Examples

			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

Crossrefs

Cf. A211790.

Programs

  • Mathematica
    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 *)

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

Views

Author

Clark Kimberling, Apr 22 2012

Keywords

Comments

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

Links

Crossrefs

Cf. A211790, A211808.

Programs

  • Maple
    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.)

A211806 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^2<=x^2+y^2.

Original entry on oeis.org

1, 5, 16, 36, 69, 119, 190, 282, 399, 547, 726, 940, 1195, 1493, 1834, 2224, 2669, 3165, 3720, 4338, 5021, 5771, 6596, 7494, 8467, 9521, 10662, 11890, 13207, 14621, 16134, 17742, 19457, 21283, 23214, 25258, 27421, 29703, 32108, 34638
Offset: 1

Views

Author

Clark Kimberling, Apr 22 2012

Keywords

Comments

Row 2 of A211808; see A211790 for a discussion and guide to related sequences.

Crossrefs

Cf. A211790, A211808.

Programs

  • Mathematica
    (See the program at A211808.)
Showing 1-4 of 4 results.

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