A361975 -id:A361975 - cp's OEIS frontend

A361993 (2,1)-block array, B(2,1), of the Wythoff array (A035513), read by descending antidiagonals.

5, 9, 15, 14, 25, 26, 23, 40, 43, 36, 37, 65, 69, 59, 47, 60, 105, 112, 95, 77, 57, 97, 170, 181, 154, 124, 93, 68, 157, 275, 293, 249, 201, 150, 111, 78, 254, 445, 474, 403, 325, 243, 179, 127, 89, 411, 720, 767, 652, 526, 393, 290, 205, 145, 99, 665, 1165

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(2,1) is a row-splitting array. The rows of B(2,1) are linearly recurrent with signature (1,1); the columns are linearly recurrent with signature (1,1,-1). The order array (as defined in A333029) of B(2,1) is A361995.

			Corner of B(2,1):
   5    9   14   23   37   60   97  157 ...
  15   25   40   65  105  170  275  445 ...
  26   43   69  112  181  293  474  767 ...
  36   59   95  154  249  403  652 1055 ...
  47   77  124  202  325  526  851 1377 ...
  ...
(column 1 of A035513) = (1,4,6,9,12,14,17,19,...), so (column 1 of B(2,1)) = (5,15,26,36,...);
(column 2 of A000027) = (2,7,10,15,20,23,28,31,...), so (column 2 of B(2,1)) = (9,25,43,59,...).

Cf. A000045, A001622, A035513, A080164, A361975, A361992 (array B(1,2)), 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[2 h - 1, k] + w[2 h, k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A361993 sequence *)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* A361993 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 Wythoff array (A035513).

b(i,j) = F(j+1) ([2 i r] + [(2 i - 1) r]) + (4 i - 3) F(j), where F = A000045, the Fibonacci numbers, and r = (1+sqrt(5))/2, the golden ratio, A001622, and [ ] = floor.

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

Original entry on oeis.org

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.

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

Original entry on oeis.org

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.

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A361993 (2,1)-block array, B(2,1), of the Wythoff array (A035513), read by descending antidiagonals.

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

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

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula