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A335499 1st Lucas-Wythoff array (w(n,k)), by antidiagonals; see Comments. 2
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, 20, 23, 29, 65, 58, 54, 44, 32, 24, 26, 47, 105, 94, 87, 72, 51, 39, 27, 30, 76, 170, 152, 141, 116, 83, 62, 43, 31, 34, 123, 275, 246, 228, 188, 134, 101, 69, 50 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

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.

LINKS

Table of n, a(n) for n=1..64.

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

FORMULA

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

column 1:  w(n,1) = u(n,1);

column 2:  w(n,2) = k + [r*[r*n]];

column k, for k >=3: w(n,k) = u(n,k-1).

EXAMPLE

Corner:

   1    2    3    4    7   11   18   29   47

   5    6   10   15   25   40   65  105  170

   8    9   14   22   36   58   94  152  246

  12   13   21   33   54   87  141  238  369

  16   17   28   44   72  116  188  304  492

  19   20   32   51   83  134  217  351  568

MATHEMATICA

r = GoldenRatio; u[n_, k_] := LucasL[k] Floor[n*r] + (n - 1) LucasL[k - 1];

v[k_] := k + Floor[r*Floor[r*k]];  (* column 2 *)

w[n_, 2] := v[n]; w[n_, k_] := u[n, k - 1]; w[n_, 1] := u[n, 1];

TableForm[Table[w[n, k], {n, 1, 15}, {k, 1, 20}]] (* A335499, array **)

Table[w[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* A335499, sequence *)

CROSSREFS

Cf. A000032, A000045, A001622, A035513, A335500.

Sequence in context: A222072 A246007 A256997 * A239970 A111202 A194280

Adjacent sequences:  A335496 A335497 A335498 * A335500 A335501 A335502

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 12 2020

STATUS

approved

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Last modified August 14 08:08 EDT 2020. Contains 336480 sequences. (Running on oeis4.)