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.

A309555 Triangle read by rows: T(n,k) = 3 + k*(n-k) for n >= 0, 0 <= k <= n.

Original entry on oeis.org

3, 3, 3, 3, 4, 3, 3, 5, 5, 3, 3, 6, 7, 6, 3, 3, 7, 9, 9, 7, 3, 3, 8, 11, 12, 11, 8, 3, 3, 9, 13, 15, 15, 13, 9, 3, 3, 10, 15, 18, 19, 18, 15, 10, 3, 3, 11, 17, 21, 23, 23, 21, 17, 11, 3, 3, 12, 19, 24, 27, 28, 27, 24, 19, 12, 3, 3, 13, 21, 27, 31, 33, 33, 31, 27, 21, 13, 3, 3, 14, 23, 30, 35, 38, 39, 38, 35, 30, 23, 14, 3
Offset: 0

Views

Author

Philip K Hotchkiss, Aug 07 2019

Keywords

Comments

The rascal triangle (A077028) can be generated by either of the rules South = (East*West+1)/North or South = East+West+1-North; this number triangle can be generated by either of the rules South = (East*West+3)/North or South = East+West+1-North.
It is more suggestive to observe that N*S-E*W = 1 or 3 in the two cases, and (N+S)-(E+W) = 1 in both cases. In fact "3" in the present definition can be replaced by any integer c, and we get a triangle of integers with N*S-E*W = c and (N+S)-(E+W) = 1. I say "suggestive", because these rules also arise in frieze patterns. - N. J. A. Sloane, Aug 28 2019

Examples

			For the row n=3: a(3,0)=3, a(3,1)=5, a(3,2)=5, a(3,3)=3, ...
For the antidiagonal r=2: T(2,0)=3, T(2,1)=5, T(2,3)=7, T(2,4)=9, ...
The triangle begins:
..............3..
............3..3..
..........3..4..3..
........3..5...5..3..
......3..6...7...6..3..
....3..7...9...9..7..3..
..3..8..11..12..11..8..3..
3..9..13..15..15..13..9..3.
...

Links

Crossrefs

Cf. A077028, A309555, A309557.

Programs

  • Maple
    T:= proc(n, k)
    if n<0 or k<0 or k>n then
       0;
    else
       k*(n-k)+3 ;
    end if;
    end:
seq(seq(T(n,k), k=0..n), n=0..12);
  • Mathematica
    T[n,k]:=k(n-k)+3;T[0,0] = 3; Table[T[n,k],{n,0,12},{k,0,n}]//Flatten

Formula

By rows: a(n,k) = 3 + k(n-k), n >= 0, 0 <= k <= n.
By antidiagonals: T(r,k) = 3 + r*k, r,k >= 0.

Extensions

Missing a(50)=23 inserted by Georg Fischer, Nov 08 2021

A309559 Triangle read by rows: T(n,k) = 1 + n + k^2/2 - k/2 + k*(n-k), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 6, 7, 7, 5, 8, 10, 11, 11, 6, 10, 13, 15, 16, 16, 7, 12, 16, 19, 21, 22, 22, 8, 14, 19, 23, 26, 28, 29, 29, 9, 16, 22, 27, 31, 34, 36, 37, 37, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 11, 20, 28, 35, 41, 46, 50, 53, 55, 56, 56, 12, 22, 31, 39, 46, 52, 57, 61, 64, 66, 67, 67, 13, 24, 34, 43, 51, 58, 64, 69, 73, 76, 78, 79, 79
Offset: 0

Views

Author

Philip K Hotchkiss, Aug 07 2019

Keywords

Comments

The rascal triangle (A077028) can be generated by the rule South = (East*West+1)/North or South = East+West+1-North; this number triangle can also be generated by South = East+West+1-North, but there not by an equation of the form South = (East*West+d)/North.

Examples

			For row n=3: T(3,0)=4, T(3,1)=6, T(3,2)=6, T(3,3)=7.
Triangle T begins:
                  1
                2   2
              3   4   4
            4   6   7   7
          5   8  10  11  11
        6  10  13  15  16  16
      7  12  16  19  21  22  22
    8  14  19  23  26  28  29  29
  9  16  22  27  31  34  36  37  37
                 ...

Links

Crossrefs

Cf. A077028, A309555, A309557.

Programs

  • Maple
    T := proc(n, k)
   if n<0 or k<0 or k>n then
       0;
   else
       1+n+(1/2)*k^2-(1/2)k +k*(n-k);
   end if;
  • Mathematica
    T[n_,k_]:=1+n+(1/2)*k^2-(1/2)k +k*(n-k); Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
f[n_] := Table[SeriesCoefficient[(-1+(3-2*x)*y+(-1+x)*y^2)/((-1+x)^2*(-1+y)^3), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Flatten[Array[f, 13,0]] (* Stefano Spezia, Sep 08 2019 *)

Formula

G.f.: (-1+(3-2*x)*y+(-1+x)*y^2)/((-1+x)^2*(-1+y)^3). - Stefano Spezia, Sep 08 2019

A332790 Triangle read by rows: T(n,k) = 1 + 2*n + k + 5*k(n-k) for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 3, 4, 5, 11, 7, 7, 18, 19, 10, 9, 25, 31, 27, 13, 11, 32, 43, 44, 35, 16, 13, 39, 55, 61, 57, 43, 19, 15, 46, 67, 78, 79, 70, 51, 22, 17, 53, 79, 95, 101, 97, 83, 59, 25, 19, 60, 91, 112, 123, 124, 115, 96, 67, 28, 21, 67, 103, 129, 145, 151, 147, 133, 109, 75, 31
Offset: 0

Views

Author

Philip K Hotchkiss, Mar 04 2020

Keywords

Examples

			From _Jon E. Schoenfield_, Mar 14 2020: (Start)
.
  n\k|  0    1    2    3    4    5    6    7    8    9   10
  ---+-----------------------------------------------------
   0 |  1
   1 |  3    4
   2 |  5   11    7
   3 |  7   18   19   10
   4 |  9   25   31   27   13
   5 | 11   32   43   44   35   16
   6 | 13   39   55   61   57   43   19
   7 | 15   46   67   78   79   70   51   22
   8 | 17   53   79   95  101   97   83   59   25
   9 | 19   60   91  112  123  124  115   96   67   28
  10 | 21   67  103  129  145  151  147  133  109   75   31
  ...
(End)

Links

Crossrefs

Cf. A077028, A309555, A309557, A309559, A332963

Programs

  • Maple
    :=proc(n, k)
   if n<0 or k<0 or k>n then
       0;
   else
       1+2*n+k+5*k*(n-k);
   end if;
  • Mathematica
    T[n_, k_]:=1+2*n+k+5*k*(n-k); Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten

Formula

T(n,k) = 1 + 2*n + k + 5*k*(n-k), n >= 0, 0 <= k <= n.

A332963 Number triangle where T(2n,0)=T(2n,2n)=1, T(2n+1,0)=T(2n+1,2n+1)=2 for all n >= 0, and the interior numbers are defined recursively by T(n,k) = (T(n-1,k-1)*T(n-1,k)+1)/T(n-2,k-1) for n > 2, 0 < k <= n.

Original entry on oeis.org

1, 2, 2, 1, 5, 1, 2, 3, 3, 2, 1, 7, 2, 7, 1, 2, 4, 5, 5, 4, 2, 1, 9, 3, 13, 3, 9, 1, 2, 5, 7, 8, 8, 7, 5, 2, 1, 11, 4, 19, 5, 19, 4, 11, 1, 2, 6, 9, 11, 12, 12, 11, 9, 6, 2, 1, 13, 5, 25, 7, 29, 7, 25, 5, 13, 1, 2, 7, 11, 14, 16, 17, 17, 16, 14, 11, 7, 2
Offset: 0

Views

Author

Philip K Hotchkiss, Mar 04 2020

Keywords

Examples

			For row 3: a(3,0)=2, a(3,1)= 3, a(3,2)=3, a(3,3)=2.
For antidiagonal 3: T(3,0)=2, T(3,1)=7, T(3,2)=5, T(3,3)=13, ...
Triangle begins:
  1;
  2, 2;
  1, 5, 1;
  2, 3, 3, 2;
  1, 7, 2, 7, 1;
  2, 4, 5, 5, 4, 2;
  ...

Links

Crossrefs

Cf. A077028, A309555, A309557, A332790, A309559

Programs

  • PARI
    T(n, k) = if ((n<0) || (nMichel Marcus, Mar 16 2020

Formula

By rows: a(2n,0)=a(2n,2n)=1, a(2n+1,0)=a(2n+1,2n+1)=2 for all n >= 0, while the interior numbers are defined recursively by a(n,k) = (a(n-1,k-1)*a(n-1,k)+1)/a(n-2,k-1) for n >= 2, 0 < k <= n.
By antidiagonals: T(0,2n)=T(2n,0)=1, T(0,2n+1)=T(2n+1,0)=2 for all n >= 0, while the interior numbers are defined recursively by T(r,k) = (T(r-1,k)*(Tr,k-1)+1)/T(r-1,k-1) for r,k > 0.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.