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A182801
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Joint-rank array of the numbers j*r^(i-1), where r = golden ratio = (1+sqrt(5))/2, i>=1, j>=1, read by antidiagonals.
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30
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1, 3, 2, 5, 6, 4, 7, 9, 11, 8, 10, 13, 16, 19, 14, 12, 18, 23, 28, 32, 25, 15, 21, 31, 39, 48, 54, 42, 17, 26, 36, 52, 66, 81, 89, 71, 20, 29, 44, 61, 86, 110, 134, 147, 117, 22, 34, 49, 73, 102, 141, 181, 221, 240, 193, 24, 38, 57, 82
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OFFSET
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1,2
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COMMENTS
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Joint-rank arrays are introduced here as follows.
Suppose that R={f(i,j)} is set of positive numbers, where i and j range through countable sets I and J, respectively, such that for every n, then number f(i,j) < n is finite. Let T(i,j) be the position of f(i,j) in the joint ranking of all the numbers in R. The joint-rank array of R is the array T whose i-th row is T(i,j).
For A182801, f(i,j)=j*r^(i-1), where r=(1+sqrt(5))/2 and I=J={1,2,3,...}.
Every positive integer occurs exactly once in A182801, so that as a sequence it is a permutation of the positive integers.
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LINKS
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FORMULA
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T(i,j)=Sum{floor(j*r^(i-k)): k>=1}.
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EXAMPLE
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Northwest corner:
1....3....5....7...10...12...
2....6....9...13...18...21...
4...11...16...23...31...36...
8...19...28...39...52...61...
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MATHEMATICA
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r=GoldenRatio;
f[i_, j_]:=Sum[Floor[j*r^(i-k)], {k, 1, i+Log[r, j]}];
TableForm[Table[f[i, j], {i, 1, 16}, {j, 1, 16}]] (* A182801 *)
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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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