A353658 Rectangular array by antidiagonals: row k lists the numbers whose Fibonacci-Lucas representation has k terms.

1, 2, 4, 3, 6, 7, 5, 9, 10, 49, 8, 11, 15, 51, 80, 13, 12, 18, 70, 83, 549, 21, 14, 19, 72, 114, 551, 889, 34, 16, 23, 77, 117, 570, 892, 6094, 55, 17, 26, 79, 125, 572, 923, 6096, 9861, 89, 20, 27, 82, 128, 782, 926, 6115, 9864, 67589

Offset: 1

Author

Clark Kimberling, May 04 2022

Keywords

• nonn
• tabl

Comments

The Fibonacci-Lucas representation of n, denoted by FL(n), is defined for n >= 1 as the sum t(1) + t(2) + ... + t(k), where t(1) is the greatest Fibonacci number (A000045(n), with n >= 2) that is <= n, and t(2) is the greatest Lucas number (A000032(n), with n >= 1) that is <= n - t(1), and so on; that is, the greedy algorithm is applied to find successive greatest Fibonacci and Lucas numbers, in alternating order, with sum n. Every positive integer occurs exactly once in the array.

Examples

			Northwest corner:
     1     2     3     5     8    13    21    34
     4     6     9    11    12    14    16    17
     7    10    15    18    19    23    26    27
    49    51    70    72    77    79    82    88
    80    83   114   117   125   128   133   143
   549   551   570   572   782   784   803   805
   889   892   923   926  1266  1269  1300  1303
  6094  6096  6115  6117  6327  6329  6348  6350
		

Crossrefs

Cf. A000032, A000045, A007895, A116543, A353655, A353656, A353659.

Programs

Mathematica

fib = Map[Fibonacci, Range[2, 51]];
luc = Map[LucasL, Range[1, 50]];
t = Map[(n = #; fl = {}; f = 0; l = 0;
     While[IntegerQ[l], n = n - f - l;
      f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n &] - 1]];
      l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n - f &] - 1]];
      AppendTo[fl, {f, l}]];
     {Total[#], #} &[Select[Flatten[fl], IntegerQ]]) &, Range[8000]];
Length[t];
u = Table[Length[t[[n]][[2]]], {n, 1, Length[t]}];
Take[u, 150]
TableForm[Table[Flatten[Position[u, k]], {k, 1, 8}]]
w[k_, n_] := Flatten[Position[u, k]][[n]]
Table[w[n - k + 1, k], {n, 8}, {k, n, 1, -1}] // Flatten
(* Peter J. C. Moses, May 04 2022 *)