A335500 - OEIS

A335500

2nd Lucas-Wythoff array (w(n,k)), by antidiagonals; see Comments.

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.

Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.

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