A353659 Rectangular array read by downwards antidiagonals: row k lists the numbers whose Lucas-Fibonacci representation has k terms.

1, 3, 2, 4, 5, 15, 7, 6, 17, 25, 11, 8, 22, 28, 172, 18, 9, 24, 36, 174, 279, 29, 10, 27, 39, 193, 282, 1913, 47, 12, 33, 44, 195, 313, 1915, 3096, 76, 13, 35, 54, 248, 316, 1934, 3099, 21221, 123, 14, 38, 57, 250, 402, 1936, 3130, 21223, 34337

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..55.

Example

Northwest corner:
     1     3     4     7    11    18    29    47    76   123
     2     5     6     8     9    10    12    13    14    16
    15    17    22    24    27    33    35    38    40    41
    25    28    36    39    44    54    57    62    65    66
   172   174   193   195   248   250   269   271   276   278
   279   282   313   316   402   405   436   439   447   450
  1913  1915  1934  1936  2146  2148  2167  2169  2756  2758
  3096  3099  3130  3133  3473  3476  3507  3510  4460  4463

Mathematica

fib = Map[Fibonacci, Range[2, 51]];
luc = Map[LucasL, Range[1, 50]];
t = Map[(n = #; lf = {}; f = 0; l = 0;
     While[IntegerQ[f], n = n - l - f;
      l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n &] - 1]];
      f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n - l &] - 1]];
      AppendTo[lf, {l, f}]];
     {Total[#], #} &[Select[Flatten[lf], IntegerQ]]) &, Range[50000]];
Length[t];
u = Table[Length[t[[n]][[2]]], {n, 1, Length[t]}];
Take[u, 150]
TableForm[Table[Flatten[Position[u, k]], {k, 1, 11}]];
w[k_, n_] := Flatten[Position[u, k]][[n]]
Table[w[n - k + 1, k], {n, 11}, {k, n, 1, -1}] // Flatten
(* Peter J. C. Moses, May 04 2022 *)

Crossrefs

Cf. A000032, A000045, A007895, A116543, A353655, A353656, A353658.

Sequence in context: A255167 A264993 A280318 * A246982 A258215 A319073

Adjacent sequences: A353656 A353657 A353658 * A353660 A353661 A353662

Keyword

nonn,tabl

Author

Clark Kimberling, May 04 2022

Status

approved