A361974 -id:A361974 - cp's OEIS frontend

A361976 (2,2)-block array, B(2,2), of the natural number array (A000027), read by descending antidiagonals.

11, 31, 39, 67, 75, 83, 119, 127, 135, 143, 187, 195, 203, 211, 219, 271, 279, 287, 295, 303, 311, 371, 379, 387, 395, 403, 411, 419, 487, 495, 503, 511, 519, 527, 535, 543, 619, 627, 635, 643, 651, 659, 667, 675, 683, 767, 775, 783, 791, 799, 807, 815, 823

Offset: 1

Clark Kimberling, Apr 01 2023

We begin with a definition. Suppose that W = (w(i,j)), where i >= 1 and j >= 1, is an array of numbers such that if m and n satisfy 1 <= m < n, then there exists k such that w(m,k+h) < w(n,h+1) < w(m,k+h+1) for every h >= 0 . Then W is a row-splitting array. The array B(2,2) is a row-splitting array. The rows and columns of B(2,2) are linearly recurrent with signature (3,-3,1). It appears that the order array (as defined in A333029) of B(2,2) is given by A000027.

			Corner of B(2,2):
   11   31   67   119   187   271
   39   75  127   195   279   379
   83  135  203   287   387   503
  143  211  295   395   511   643
  219  303  403   519   651   799

Cf. A000027, A333029, A361974 (array B(1,2)), A361975 (array B(2,1)).

zz = 10; z = 13;
w[n_, k_] := n + (n + k - 2) (n + k - 1)/2;
t[n_, k_] := w[2 n - 1, 2 k - 1] + w[2 n - 1, 2 k] + w[2 n, 2 k - 1] + w[2 n, 2 k]
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (*A361976 sequence*)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (*A361976 array*)

B(2,2) = (b(i,j)), where b(i,j) = w(2i-1,2j-1) + w(2i-1,2j) + w(2i,2j-1) + w(2i, 2j) for i >= 1, j >=1, where (w(i,j)) is the natural number array (A000027).

b(i,j) = 8(i+j)^2 - 12i - 20 j + 11.

A361975 (2,1)-block array, B(2,1), of the natural number array (A000027), read by descending antidiagonals.

Original entry on oeis.org

4, 7, 16, 12, 23, 36, 19, 32, 47, 64, 28, 43, 60, 79, 100, 39, 56, 75, 96, 119, 144, 52, 71, 92, 115, 140, 167, 196, 67, 88, 111, 136, 163, 192, 223, 256, 84, 107, 132, 159, 188, 219, 252, 287, 324, 103, 128, 155, 184, 215, 248, 283, 320, 359, 400, 124, 151

Offset: 1

Clark Kimberling, Apr 01 2023

We begin with a definition. Suppose that W = (w(i,j)), where i >= 1 and j >= 1, is an array of numbers such that if m and n satisfy 1 <= m < n, then there exists k such that w(m,k+h) < w(n,h+1) < w(m,k+h+1) for every h >= 0. Then W is a row-splitting array. The array B(2,1) is a row-splitting array. The rows and columns of B(2,1) are linearly recurrent with signature (3,-3,1). It appears that the order array (as defined in A333029) of B(2,1) is given by A056537 (the dispersion of the nonsquares).

			Corner of B(2,1):
    4    7   12   19   28   39   52
   16   23   32   43   56   71   88
   36   47   60   75   92  111  132
   64   79   96  115  136  159  184
  100  119  140  163  188  215  244
  144  167  192  219  238  279  312
(column 1 of A000027) = (1,3,6,10,15,21,...), so (column 1 of B(2,1)) = (4,16,64,...);
(column 2 of A000027) = (2,5,9,14,20,27,...), so (column 2 of B(2,1)) = (7,23,47,...).

Cf. A000027, A056537, A333029, A361974 (array B(1,2)), A361976 (array B(2,2)).

zz = 10; z = 13;
w[n_, k_] := n + (n + k - 2) (n + k - 1)/2;
t[h_, k_] := w[2 h - 1, k] + w[2 h, k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* this sequence *)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* this sequence as an array *)

B(2,1) = (b(i,j)), where b(i,j) = w(2i-1, j) + w(2i, j) for i >= 1, j >= 1, where (w(i,j)) is the natural number array (A000027).

b(i,j) = 4i - 1 + (2i + j - 2)^2.

A361992 (1,2)-block array, B(1,2), of the Wythoff array (A035513), read by descending antidiagonals.

Original entry on oeis.org

3, 8, 11, 21, 29, 16, 55, 76, 42, 24, 144, 199, 110, 63, 32, 377, 521, 288, 165, 84, 37, 987, 1364, 754, 432, 220, 97, 45, 2584, 3571, 1974, 1131, 576, 254, 118, 50, 6765, 9349, 5168, 2961, 1508, 665, 309, 131, 58, 17711, 24476, 13530, 7752, 3948, 1741, 809

Offset: 1

Clark Kimberling, Apr 04 2023

We begin with a definition. Suppose that W = (w(i,j)), where i >= 1 and j >= 1, is an array of numbers such that if m and n satisfy 1 <= m < n, then there exists k such that w(m,k+h) < w(n,h+1) < w(m,k+h+1) for every h >= 0. Then W is a row-splitting array. The array B(1,2) is a row-splitting array. The rows of B(1,2) are linearly recurrent with signature (3,-1). The order array (as defined in A333029) of B(1,2) is the Wythoff difference array, A080164.

			Corner of B(1,2):
   3     8    21    55   144   377   987 ...
  11    29    76   199   521  1364  3571 ...
  16    42   110   288   754  1974  5168 ...
  24    63   165   432  1131  2961  7752 ...
  32    84   220   576  1508  3948 10336 ...
  ...
(row 1 of A035513) = (1,2,3,5,8,13,21,34,...), so (row 1 of B(1,2)) = (3,8,21,55,...);
(row 2 of A000027) = (4,7,11,18,29,47,76,123,...), so (row 2 of B(1,2)) = (11,29,76,199,...).

Cf. A000045, A001622, A035513, A080164, A361974, A361993 (array B(2,1)), A361994 (array B(2,2)).

f[n_] := Fibonacci[n]; r = GoldenRatio;
zz = 10; z = 13;
w[n_, k_] := f[k + 1] Floor[n*r] + (n - 1) f[k]
t[h_, k_] := w[h, 2 k - 1] + w[h, 2 k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (* A361992 sequence *)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* A361992 array *)

B(1,2) = (b(i,j)), where b(i,j) = w(i, 2j-1) + w(i, 2j) for i >= 1, j >= 1, where (w(i,j)) is the Wythoff array (A035513).

b(i,j) = w(i,2j+1) = F(2 k + 2)*floor(h r) + (h - 1)F(2 k + 1), where F = A000045, the Fibonacci numbers, and r = (1+sqrt(5))/2, the golden ratio, A001622.

cp's OEIS Frontend

A361976 (2,2)-block array, B(2,2), of the natural number array (A000027), read by descending antidiagonals.

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A361975 (2,1)-block array, B(2,1), of the natural number array (A000027), read by descending antidiagonals.

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A361992 (1,2)-block array, B(1,2), of the Wythoff array (A035513), read by descending antidiagonals.

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