The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 15 17:11 EDT 2021. Contains 343920 sequences. (Running on oeis4.)