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A254105 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), ...

%I A254105 #25 Oct 21 2024 01:20:32
%S A254105 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 A254105 36,21,11,248,249,185,123,75,44,24,15,503,504,376,250,154,91,51,30,16,
%U A254105 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 A254105 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 A254105 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 A254105 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 A254105 For other examples of such sequences, see the Crossrefs section. For a general introduction, please follow the Kimberling references.
%C A254105 The main diagonal: 1, 6, 20, 60, 154, 377, 887, 2040, 4598, 10229, 22515, 49139, ...
%H A254105 Antti Karttunen, <a href="/A254105/b254105.txt">Table of n, a(n) for n = 1..120; the first 15 antidiagonals of array</a>
%H A254105 Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions and Dispersions</a>.
%H A254105 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 A254105 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A254105 If col = 1, then A(row,col) = A005187(row), otherwise A(row,col) = A055938(A(row,col-1)).
%e A254105 The top left corner of the array:
%e A254105    1,  2,  5,  12,  27,  58,  121,  248,  503,  1014,  2037,  4084
%e A254105    3,  6, 13,  28,  59, 122,  249,  504, 1015,  2038,  4085,  8180
%e A254105    4,  9, 20,  43,  90, 185,  376,  759, 1526,  3061,  6132, 12275
%e A254105    7, 14, 29,  60, 123, 250,  505, 1016, 2039,  4086,  8181, 16372
%e A254105    8, 17, 36,  75, 154, 313,  632, 1271, 2550,  5109, 10228, 20467
%e A254105   10, 21, 44,  91, 186, 377,  760, 1527, 3062,  6133, 12276, 24563
%e A254105   11, 24, 51, 106, 217, 440,  887, 1782, 3573,  7156, 14323, 28658
%e A254105   15, 30, 61, 124, 251, 506, 1017, 2040, 4087,  8182, 16373, 32756
%e A254105   16, 33, 68, 139, 282, 569, 1144, 2295, 4598,  9205, 18420, 36851
%e A254105   18, 37, 76, 155, 314, 633, 1272, 2551, 5110, 10229, 20468, 40947
%e A254105 etc.
%o A254105 (Scheme)
%o A254105 (define (A254105 n) (A254105bi (A002260 n) (A004736 n)))
%o A254105 (define (A254105bi row col) (if (= 1 col) (A005187 row) (A055938 (A254105bi row (- col 1)))))
%Y A254105 Inverse: A254106.
%Y A254105 Transpose: A254107.
%Y A254105 Column 1: A005187.
%Y A254105 Cf. also A000325, A095768, A123720 (Seem to be rows 1 - 3, the last one from its second term onward.)
%Y A254105 Columnd index of n: A254111, Row index: A254112.
%Y A254105 Cf. A002260, A004736, A055938, A233275-A233278.
%Y A254105 Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191450, A191426-A191455.
%K A254105 nonn,tabl
%O A254105 1,2
%A A254105 _Antti Karttunen_, Jan 26 2015