A367832 Array T(n, k) read by ascending antidiagonals is a dispersion based on A367467. Column 1 lists the numbers which cannot be represented by A367467(m) + m. For k >= 1, T(n, k+1) = A367467(T(n, k)) + T(n, k).

1, 4, 2, 7, 6, 3, 11, 9, 10, 5, 14, 12, 15, 17, 8, 18, 16, 20, 25, 29, 13, 21, 19, 27, 34, 42, 49, 22, 24, 23, 32, 46, 58, 71, 83, 37, 28, 26, 39, 54, 78, 99, 121, 141, 63, 31, 30, 44, 66, 92, 133, 169, 206, 240, 107, 35, 33, 51, 75, 112, 157, 227, 288, 351, 409, 182, 38, 36, 56, 87, 128, 191, 268

Offset: 1

Author

Thomas Scheuerle, Dec 02 2023

Keywords

• nonn
• tabl

Comments

This sequence is a permutation of the positive integers.

The array T(n, k+1) - T(n, k) for k > 1 is also a permutation of the positive integers.

Columns k > 2 together consist of all the numbers from A003152. These are all the positive numbers of the form floor(m*(1+1/sqrt(2))).

In column 2 are all the numbers from A184119. These are all the numbers of the form floor((2+sqrt(2))*m - sqrt(2)/2).

Column 2 together with the columns k > 2 are all the numbers from A087057; these are all the numbers of the form ceiling(m*sqrt(2)). Together with column 1, which consists of all the numbers from A083051, they cover all positive integers.

An alternative definition that allows this array to be obtained without using A367467:

Take for T(n, 1) and T(n, 2) the first and the second number which do not appear in any row r < n. Complete all rows by the recurrence T(n, k) = floor(T(n, k-1)*(1 + 1/sqrt(2))). Start in the first row with T(1, 1) = 1 and T(1, 2) = 2.

Let Q(n, k) = T(n, k+2) - T(n, k+1) for k > 0. Let b(m) be the row n where the integer m is found in Q. Then we will obtain for (b(n)) the sequence: 1, 1, 1, 2, 1, 3, 2, 4, 1, 3, 5, 2, 6, 4, 1, ... . If we were to remove the first occurrence of each number in this sequence, we would get the same sequence again, hence (b(n)) is a fractal sequence.

Examples

			Array T(n, k) begins:
   1,  2,  3,  5,   8,  13,  22,  37,  63,  107, ...
   4,  6, 10, 17,  29,  49,  83, 141, 240,  409, ...
   7,  9, 15, 25,  42,  71, 121, 206, 351,  599, ...
  11, 12, 20, 34,  56,  99, 169, 288, 491,  839, ...
  14, 16, 27, 46,  78, 133, 227, 387, 660, 1126, ...
  18, 19, 32, 54,  92, 157, 268, 457, 780, 1331, ...
  21, 23, 39, 66, 112, 191, 326, 556, 949, 1620, ...
  ...
		

References

• Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157-168.

Links

• Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A083051, A087057, A184119, A293078, A328987, A367467.

Cf. A083050 (a closely related dispersion).

Formula

T(1, k) = A293078(k).

T(n, 1) = A083051(n-1).

T(n, 2) = A184119(n).

Conjectured: T(n, 3) = A328987(n-1).

T(1, k) = 2*T(1, k-1) - T(1, k-2) + floor(T(1, k-2)/2), for k > 2.

T(n, k+1) = floor(T(n, k)*(1+1/sqrt(2))) for k > 1.

T(n, k+1) = A367467(T(n, k)) + T(n, k).

Extensions

Edited by Peter Munn, Dec 11 2023