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A335500
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2nd Lucas-Wythoff array (w(n,k)), by antidiagonals; see Comments.
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2
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1, 3, 5, 4, 10, 8, 7, 15, 14, 12, 11, 25, 22, 21, 16, 18, 40, 36, 33, 28, 19, 29, 65, 58, 54, 44, 32, 23, 47, 105, 94, 87, 72, 51, 39, 26, 76, 170, 152, 141, 116, 83, 62, 43, 30, 123, 275, 246, 228, 188, 134, 101, 69, 50, 34, 199, 445, 398, 369, 304, 217
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OFFSET
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1,2
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COMMENTS
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Let (L(n)) be the Lucas sequecce, A000032. Every positive integer n is a unique sum of distinct nonconsecutive Lucas numbers as given by the greedy algorithm. Let m(n) be the least term in this representation. Column k of the array shows the numbers n having m(n) = L(k), for k >= 1. The array is comparable to the Wythoff array, A035513, in which column k shows the numbers whose Zeckendorf representation (a sum of nonconsecutive Fibonacci numbers, A000045) has least term F(k+2), and every row satisfies the Fibonacci recurrence. Missing are the numbers n for which the least term of the Lucas representation of n is L(0) = 2. The result of inserting these numbers as a second column is the 1st Lucas-Wythoff array, A335499.
The order array of the 2nd Lucas-Wythoff array, formed by replacing each w(n,k) by its position, or rank, when all the numbers w(n,k) are arranged in increasing order, is the Wythoff array.
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LINKS
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L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Lucas representations, Fibonacci Quart. 10 (1972), 29-42, 70, 112.
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FORMULA
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Define w(n,k) = [n*r]L(k) + (n-1)L(k-1), where L = A000032 (Lucas numbers), r = golden ratio (A001622) and [ ] = floor.
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EXAMPLE
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Corner:
1 3 4 7 11 18 29 47
5 10 15 25 40 65 105 170
8 14 22 36 58 94 152 246
12 21 33 54 87 141 238 369
16 28 44 72 116 188 304 492
19 32 51 83 134 217 351 568
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MATHEMATICA
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r = GoldenRatio; LL[n_, k_] := Floor[n*r] LucasL[k] + (n - 1) LucasL[k - 1];
TableForm[Table[LL[n, k], {n, 1, 15}, {k, 1, 10}]] (* A335500, array *)
Table[LL[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A335500, sequence *)
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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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