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A254105
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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), ...
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9
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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, 36, 21, 11, 248, 249, 185, 123, 75, 44, 24, 15, 503, 504, 376, 250, 154, 91, 51, 30, 16, 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
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OFFSET
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1,2
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COMMENTS
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This sequence is one instance of Clark Kimberling's generic dispersion arrays. Paraphrasing his explanation in A191450, mutatis mutandis, we have the following definition:
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).
For other examples of such sequences, see the Crossrefs section. For a general introduction, please follow the Kimberling references.
The main diagonal: 1, 6, 20, 60, 154, 377, 887, 2040, 4598, 10229, 22515, 49139, ...
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LINKS
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FORMULA
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If col = 1, then A(row,col) = A005187(row), otherwise A(row,col) = A055938(A(row,col-1)).
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EXAMPLE
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The top left corner of the array:
1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084
3, 6, 13, 28, 59, 122, 249, 504, 1015, 2038, 4085, 8180
4, 9, 20, 43, 90, 185, 376, 759, 1526, 3061, 6132, 12275
7, 14, 29, 60, 123, 250, 505, 1016, 2039, 4086, 8181, 16372
8, 17, 36, 75, 154, 313, 632, 1271, 2550, 5109, 10228, 20467
10, 21, 44, 91, 186, 377, 760, 1527, 3062, 6133, 12276, 24563
11, 24, 51, 106, 217, 440, 887, 1782, 3573, 7156, 14323, 28658
15, 30, 61, 124, 251, 506, 1017, 2040, 4087, 8182, 16373, 32756
16, 33, 68, 139, 282, 569, 1144, 2295, 4598, 9205, 18420, 36851
18, 37, 76, 155, 314, 633, 1272, 2551, 5110, 10229, 20468, 40947
etc.
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PROG
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(Scheme)
(define (A254105bi row col) (if (= 1 col) (A005187 row) (A055938 (A254105bi row (- col 1)))))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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