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%I #13 Feb 05 2021 18:19:12
%S 1,2,5,3,6,8,4,10,9,12,7,15,14,13,16,11,25,22,21,17,19,18,40,36,33,28,
%T 20,23,29,65,58,54,44,32,24,26,47,105,94,87,72,51,39,27,30,76,170,152,
%U 141,116,83,62,43,31,34,123,275,246,228,188,134,101,69,50
%N 1st Lucas-Wythoff array (w(n,k)), by antidiagonals; see Comments.
%C Let (L(n)) be the Lucas sequence, 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 1 of the array shows the numbers n having m(n) = L(1); column 2 shows those n having m(n) = L(0) = 2. For k >= 3, column k shows those n having m(n) = L(k). 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).
%C The 1st Lucas-Wythoff array has increasing rows and increasing columns, and every positive integer occurs exactly once. However, the Fibonacci recurrence for rows (as in the Wythoff array), does not hold. Deleting column 2 leaves the 2nd Lucas-Wythoff array (A335500), in which the Fibonacci recurrence holds for rows.
%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 u(n,k) = [n*r]L(k) + (n-1)L(k-1), where L = A000032 (Lucas numbers), r = golden ratio (A001622) and [ ] = floor. Then
%F column 1: w(n,1) = u(n,1);
%F column 2: w(n,2) = k + [r*[r*n]];
%F column k, for k >=3: w(n,k) = u(n,k-1).
%e Corner:
%e 1 2 3 4 7 11 18 29 47
%e 5 6 10 15 25 40 65 105 170
%e 8 9 14 22 36 58 94 152 246
%e 12 13 21 33 54 87 141 238 369
%e 16 17 28 44 72 116 188 304 492
%e 19 20 32 51 83 134 217 351 568
%t r = GoldenRatio; u[n_, k_] := LucasL[k] Floor[n*r] + (n - 1) LucasL[k - 1];
%t v[k_] := k + Floor[r*Floor[r*k]]; (* column 2 *)
%t w[n_, 2] := v[n]; w[n_, k_] := u[n, k - 1]; w[n_, 1] := u[n, 1];
%t TableForm[Table[w[n, k], {n, 1, 15}, {k, 1, 20}]] (* A335499, array **)
%t Table[w[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* A335499, sequence *)
%Y Cf. A000032, A000045, A001622, A035513, A335500.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 12 2020
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