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A335500 2nd Lucas-Wythoff array (w(n,k)), by antidiagonals; see Comments. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..61.

L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Lucas representations, Fibonacci Quart. 10 (1972), 29-42, 70, 112.

FORMULA

Define w(n,k) = [n*r]L(k) + (n-1)L(k-1), where L = A000032 (Lucas numbers), r = golden ratio (A001622) and [ ] = floor.

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A000032, A000045, A001622, A035513, A335499.

Sequence in context: A294673 A078439 A007063 * A127397 A284048 A326119

Adjacent sequences:  A335497 A335498 A335499 * A335501 A335502 A335503

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 12 2020

STATUS

approved

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Last modified September 21 19:57 EDT 2020. Contains 337273 sequences. (Running on oeis4.)