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A035486
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Kimberling's expulsion array read by antidiagonals.
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13
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1, 2, 2, 3, 3, 4, 4, 4, 2, 6, 5, 5, 5, 2, 8, 6, 6, 6, 7, 7, 6, 7, 7, 7, 4, 9, 2, 13, 8, 8, 8, 8, 2, 11, 12, 2, 9, 9, 9, 9, 10, 9, 8, 11, 18, 10, 10, 10, 10, 6, 12, 9, 16, 17, 16, 11, 11, 11, 11, 11, 7, 14, 14, 12, 14, 23, 12, 12, 12, 12, 12, 13, 11, 6, 9, 21, 2, 13, 13, 13, 13, 13, 13, 8, 15
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OFFSET
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1,2
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COMMENTS
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To get the next row, start with the first element to the right of the diagonal term, then take the first to the left of the diagonal, then the second to the right, then the second to the left, the third to the right, etc.
It is conjectured since 1992 that the main diagonal elements (A007063) are a permutation of the positive integers.
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REFERENCES
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R. K. Guy, Unsolved Problems Number Theory, Sect E35.
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LINKS
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Clark Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991 and Vol. 18, March 1992, p. 82-83.
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EXAMPLE
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The array starts (with elements of A007063 in brackets):
[1] 2 3 4 5 6 7 8 9 10 11 12 ...
2 [3] 4 5 6 7 8 9 10 11 12 13 ...
4 2 [5] 6 7 8 9 10 11 12 13 14 ...
6 2 7 [4] 8 9 10 11 12 13 14 15 ...
8 7 9 2 [10] 6 11 12 13 14 15 16 ...
6 2 11 9 12 [7] 13 8 14 15 16 17 ...
13 12 8 9 14 11 [15] 2 16 6 17 18 ...
2 occurs as diagonal element in row 25, 27 in row 7598, and 19 in row 49595 (cf. A006852).
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MATHEMATICA
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K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
K[i_, j_] := K[i - 1, i - (j + 2)/2] /; (EvenQ[j] && (j < 2 i - 3));
K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
K[i_] := K[i] = K[i, i]; SetAttributes[K, Listable];
T[n_] := n*(n + 1)/2;
S[n_] := Floor[1/2 (1 + Sqrt[1 + 8 (n - 1)])];
AJ[n_] := 1 + T[S[n]] - n;
AI[n_] := 1 + S[n] - AJ[n];
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PROG
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(Python)
if k >= 2*n-3: return n+k-1
q, r = divmod(k+1, 2)
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CROSSREFS
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Cf. A175312 (maximum value on lower shuffle part).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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