OFFSET
1,2
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.
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.
FORMULA
EXAMPLE
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, ...
...
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Thomas Scheuerle, Dec 02 2023
EXTENSIONS
Edited by Peter Munn, Dec 11 2023
STATUS
approved