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%I #21 Sep 09 2017 19:26:25
%S 1,2,3,5,6,4,12,13,9,7,27,28,20,14,8,58,59,43,29,17,10,121,122,90,60,
%T 36,21,11,248,249,185,123,75,44,24,15,503,504,376,250,154,91,51,30,16,
%U 1014,1015,759,505,313,186,106,61,33,18,2037,2038,1526,1016,632,377,217,124,68,37,19,4084,4085,3061,2039,1271,760,440,251,139,76,40,22
%N Dispersion of A055938; starting from its complementary sequence A005187 as the first column of square array A(row,col), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...
%C This sequence is one instance of _Clark Kimberling_'s generic dispersion arrays. Paraphrasing his explanation in A191450, mutatis mutandis, we have the following definition:
%C Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n)), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n) = {index of the row of D that contains n} is a fractal sequence. In this case s(n) = A055938(n), t(n) = A005187(n) [from term A005187(1) onward] and u(n) = A254112(n).
%C For other examples of such sequences, see the Crossrefs section. For a general introduction, please follow the Kimberling references.
%C The main diagonal: 1, 6, 20, 60, 154, 377, 887, 2040, 4598, 10229, 22515, 49139, ...
%H Antti Karttunen, <a href="/A254105/b254105.txt">Table of n, a(n) for n = 1..120; the first 15 antidiagonals of array</a>
%H Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions and Dispersions</a>.
%H Clark Kimberling, <a href="http://www.jstor.org/discover/10.2307/2159163">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 If col = 1, then A(row,col) = A005187(row), otherwise A(row,col) = A055938(A(row,col-1)).
%e The top left corner of the array:
%e 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084
%e 3, 6, 13, 28, 59, 122, 249, 504, 1015, 2038, 4085, 8180
%e 4, 9, 20, 43, 90, 185, 376, 759, 1526, 3061, 6132, 12275
%e 7, 14, 29, 60, 123, 250, 505, 1016, 2039, 4086, 8181, 16372
%e 8, 17, 36, 75, 154, 313, 632, 1271, 2550, 5109, 10228, 20467
%e 10, 21, 44, 91, 186, 377, 760, 1527, 3062, 6133, 12276, 24563
%e 11, 24, 51, 106, 217, 440, 887, 1782, 3573, 7156, 14323, 28658
%e 15, 30, 61, 124, 251, 506, 1017, 2040, 4087, 8182, 16373, 32756
%e 16, 33, 68, 139, 282, 569, 1144, 2295, 4598, 9205, 18420, 36851
%e 18, 37, 76, 155, 314, 633, 1272, 2551, 5110, 10229, 20468, 40947
%e etc.
%o (Scheme)
%o (define (A254105 n) (A254105bi (A002260 n) (A004736 n)))
%o (define (A254105bi row col) (if (= 1 col) (A005187 row) (A055938 (A254105bi row (- col 1)))))
%Y Inverse: A254106.
%Y Transpose: A254107.
%Y Column 1: A005187.
%Y Cf. also A000325, A095768, A123720 (Seem to be rows 1 - 3, the last one from its second term onward.)
%Y Columnd index of n: A254111, Row index: A254112.
%Y Cf. A002260, A004736, A055938, A233275-A233278.
%Y Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191450, A191426-A191455.
%K nonn,tabl
%O 1,2
%A _Antti Karttunen_, Jan 26 2015
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