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A254051
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Square array A by downward antidiagonals: A(n,k) = (3 + 3^n*(2*floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...
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17
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1, 3, 2, 4, 8, 5, 6, 11, 23, 14, 7, 17, 32, 68, 41, 9, 20, 50, 95, 203, 122, 10, 26, 59, 149, 284, 608, 365, 12, 29, 77, 176, 446, 851, 1823, 1094, 13, 35, 86, 230, 527, 1337, 2552, 5468, 3281, 15, 38, 104, 257, 689, 1580, 4010, 7655, 16403, 9842, 16, 44, 113, 311, 770, 2066, 4739, 12029, 22964, 49208, 29525, 18, 47
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OFFSET
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1,2
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COMMENTS
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This is transposed dispersion of (3n-1), starting from its complement A032766 as the first row of square array A(row,col). Please see the transposed array A191450 for references and background discussion about dispersions.
For any odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 -> x (A165355) is found in this array at A(row+1,col).
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LINKS
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FORMULA
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In A(n,k)-formulas below, n is the row, and k the column index, both starting from 1:
A(n,k) = (1/2) * (1 + A003961((2^(n-1)) * A254050(k))). [The above expands to this.]
A(1,k) = A032766(k) and for n > 1: A(n,k) = (3 * A254051(n-1,k)) - 1. [The definition of transposed dispersion of (3n-1).]
A(n,k) = (1+A135765(n,k))/2, or when expressed one-dimensionally, a(n) = (1+A135765(n))/2.
As a composition of related permutations. All sequences interpreted as one-dimensional:
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EXAMPLE
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The top left corner of the array:
1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21
2, 8, 11, 17, 20, 26, 29, 35, 38, 44, 47, 53, 56, 62
5, 23, 32, 50, 59, 77, 86, 104, 113, 131, 140, 158, 167, 185
14, 68, 95, 149, 176, 230, 257, 311, 338, 392, 419, 473, 500, 554
41, 203, 284, 446, 527, 689, 770, 932, 1013, 1175, 1256, 1418, 1499, 1661
...
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PROG
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(Scheme, several versions)
(define (A254051bi row col) (/ (+ 3 (* (A000244 row) (- (* 2 (A032766 col)) 1))) 6))
(define (A254051bi row col) (/ (+ 1 (* (A000244 (- row 1)) (A007310 col))) 2))
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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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