A361976 -id:A361976 - cp's OEIS frontend

A361974 (1,2)-block array, B(1,2), of the natural number array (A000027), read by descending antidiagonals.

3, 11, 8, 27, 20, 15, 51, 40, 31, 24, 83, 68, 55, 44, 35, 123, 104, 87, 72, 59, 48, 171, 148, 127, 108, 91, 76, 63, 227, 200, 175, 152, 131, 112, 95, 80, 291, 260, 231, 204, 179, 156, 135, 116, 99, 363, 328, 295, 264, 235, 208, 183, 160, 139, 120, 443, 404

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(1,2) is a row-splitting array. The rows and columns of B(1,2) are linearly recurrent with signature (3,-3,1). It appears that the order array (as defined in A333029) of B(1,2) is given by A163255.

			Corner of B(1,2):
   3   11   27   51   83  123   171   227
   8   20   40   68  104  148   200   260
  15   31   55   87  127  175   231   295
  24   44   72  108  152  204   264   332
  35   59   91  131  179  235   299   371
  48   76  112  156  298  268   336   412
(row 1 of A000027) = (1,2,4,7,11,16,22,29,...), so (row 1 of B(1,2)) = (3,11,27,58,...);
(row 2 of A000027) = (3,5,8,12,17,23,30,38,...), so (row 2 of B(1,2)) = (8,20,40,68,...).

Cf. A000027, A163255, A333029, A361975 (array B(2,1)), A361976 (array B(2,2)).

zz = 10; z = 13;
w[n_, k_] := n + (n + k - 2) (n + k - 1)/2;
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 (*A361974 sequence*)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (*A361974 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 natural number array (A000027).

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

A361994 (2,2)-block array, B(2,1), of the Wythoff array (A035513), read by descending antidiagonals.

Original entry on oeis.org

14, 37, 40, 97, 105, 69, 254, 275, 181, 95, 665, 720, 474, 249, 124, 1741, 1885, 1241, 652, 325, 150, 4558, 4935, 3249, 1707, 851, 393, 179, 11933, 12920, 8506, 4469, 2228, 1029, 469, 205, 31241, 33825, 22269, 11700, 5833, 2694, 1228, 537, 234, 81790, 88555

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(2,2) is a row-splitting array. The rows of B(2,2) are linearly recurrent with signature (3,-1); the columns are linearly recurrent with signature (1,1,-1). The order array (as defined in A333029) of B(2,2) is A361996.

			Corner of B(2,2):
   14    37    97   254   665   1741 ...
   40   105   275   720  1885   4935 ...
   69   181   474  1241  3249   8506 ...
   95   249   652  1707  4469  11700 ...
  124   325   851  2228  5833  15271 ...
  ...
b(1,1) = w(1,1) + w(1,2) + w(2,1) + w(2,2) = 1 +  2 +  4 +  7 = 14;
b(1,2) = w(1,3) + w(1,4) + w(2,3) + w(2,4) = 3 +  5 + 11 + 18 = 37;
b(2,1) = w(3,1) + w(3,2) + w(4,1) + w(4,2) = 8 + 10 +  9 + 15 = 40.

Cf. A000045, A001622, A035513, A080164, A361976, A361992 (array B(1,2)), A361993 (array B(2,1)).

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[2 h - 1, 2 k - 1] + w[2 h - 1, 2 k] + w[2 h, 2 k - 1] + w[2 h, 2 k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (*A361994 sequence *)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* A361994 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 Wythoff array (A035513).

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.

cp's OEIS Frontend

A361974 (1,2)-block array, B(1,2), of the natural number array (A000027), read by descending antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A361994 (2,2)-block array, B(2,1), of the Wythoff array (A035513), read by descending antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula