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%I #12 Feb 05 2021 18:19:24
%S 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,
%T 44,32,23,47,105,94,87,72,51,39,26,76,170,152,141,116,83,62,43,30,123,
%U 275,246,228,188,134,101,69,50,34,199,445,398,369,304,217
%N 2nd Lucas-Wythoff array (w(n,k)), by antidiagonals; see Comments.
%C 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.
%C 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.
%H L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., <a href="https://www.fq.math.ca/Scanned/10-1/carlitz2-a.pdf">Lucas representations</a>, Fibonacci Quart. 10 (1972), 29-42, 70, 112.
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Kimberling/kimber12.html">Lucas Representations of Positive Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
%F Define w(n,k) = [n*r]L(k) + (n-1)L(k-1), where L = A000032 (Lucas numbers), r = golden ratio (A001622) and [ ] = floor.
%e Corner:
%e 1 3 4 7 11 18 29 47
%e 5 10 15 25 40 65 105 170
%e 8 14 22 36 58 94 152 246
%e 12 21 33 54 87 141 238 369
%e 16 28 44 72 116 188 304 492
%e 19 32 51 83 134 217 351 568
%t r = GoldenRatio; LL[n_, k_] := Floor[n*r] LucasL[k] + (n - 1) LucasL[k - 1];
%t TableForm[Table[LL[n, k], {n, 1, 15}, {k, 1, 10}]] (* A335500, array *)
%t Table[LL[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A335500, sequence *)
%Y Cf. A000032, A000045, A001622, A035513, A335499.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 12 2020
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