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 1-5 of 5 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.

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

Views

Author

Clark Kimberling, Apr 22 2012

Keywords

Comments

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.

Examples

			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).

Links

Crossrefs

Cf. A211790, A211802.

Programs

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

Formula

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)

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

Views

Author

Clark Kimberling, Apr 22 2012

Keywords

Comments

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.

Examples

			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

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}]  (* 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 *)

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

Views

Author

Clark Kimberling, Apr 22 2012

Keywords

Comments

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

Crossrefs

Cf. A211790, A211802.

Programs

  • Mathematica
    (See the program at A211802.)

A211800 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2*w^2 < x^2 + y^2.

Original entry on oeis.org

0, 3, 13, 32, 64, 113, 181, 272, 388, 535, 713, 926, 1180, 1475, 1815, 2204, 2646, 3141, 3695, 4312, 4992, 5741, 6563, 7460, 8432, 9485, 10625, 11850, 13166, 14579, 16089, 17696, 19410, 21233, 23161, 25204, 27366, 29647, 32051, 34580
Offset: 1

Views

Author

Clark Kimberling, Apr 22 2012

Keywords

Comments

Row 2 of A211802; see A211790 for a discussion and guide to related sequences.

Crossrefs

Cf. A211790, A211801, A211802.

Programs

  • Mathematica
    (See the program at A211802.)

Extensions

Definition corrected by Georg Fischer, Sep 10 2022
Showing 1-5 of 5 results.

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