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%I #55 Dec 11 2023 21:38:39
%S 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,
%T 49,22,24,23,32,46,58,71,83,37,28,26,39,54,78,99,121,141,63,31,30,44,
%U 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
%N 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).
%C This sequence is a permutation of the positive integers.
%C The array T(n, k+1) - T(n, k) for k > 1 is also a permutation of the positive integers.
%C 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))).
%C 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).
%C 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.
%C An alternative definition that allows this array to be obtained without using A367467:
%C 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.
%C 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.
%D Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157-168.
%H Clark Kimberling, <a href="http://dx.doi.org/10.1090/S0002-9939-1993-1111434-0">Interspersions and dispersions</a>, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F T(1, k) = A293078(k).
%F T(n, 1) = A083051(n-1).
%F T(n, 2) = A184119(n).
%F Conjectured: T(n, 3) = A328987(n-1).
%F T(1, k) = 2*T(1, k-1) - T(1, k-2) + floor(T(1, k-2)/2), for k > 2.
%F T(n, k+1) = floor(T(n, k)*(1+1/sqrt(2))) for k > 1.
%F T(n, k+1) = A367467(T(n, k)) + T(n, k).
%e Array T(n, k) begins:
%e 1, 2, 3, 5, 8, 13, 22, 37, 63, 107, ...
%e 4, 6, 10, 17, 29, 49, 83, 141, 240, 409, ...
%e 7, 9, 15, 25, 42, 71, 121, 206, 351, 599, ...
%e 11, 12, 20, 34, 56, 99, 169, 288, 491, 839, ...
%e 14, 16, 27, 46, 78, 133, 227, 387, 660, 1126, ...
%e 18, 19, 32, 54, 92, 157, 268, 457, 780, 1331, ...
%e 21, 23, 39, 66, 112, 191, 326, 556, 949, 1620, ...
%e ...
%Y Cf. A083051, A087057, A184119, A293078, A328987, A367467.
%Y Cf. A083050 (a closely related dispersion).
%K nonn,tabl
%O 1,2
%A _Thomas Scheuerle_, Dec 02 2023
%E Edited by _Peter Munn_, Dec 11 2023
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