A353656 Number of terms in the Lucas-Fibonacci representation of n.

1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 3, 1, 2, 2, 2, 3, 2, 3, 4, 2, 3, 4, 1, 2, 2, 2, 3, 2, 3, 4, 2, 3, 4, 3, 3, 2, 3, 4, 3, 3, 1, 2, 2, 2, 3, 2, 3, 4, 2, 3, 4, 3, 3, 2, 3, 4, 3, 3, 4, 4, 3, 2, 3, 4, 3, 3, 4, 4, 3, 1, 2, 2, 2, 3, 2, 3, 4, 2, 3, 4

Offset: 1

Author

Clark Kimberling, May 04 2022

Keywords

• nonn

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. (See Example.)

Examples

			   n     LF(n)
   1  =  1
   2  =  1 + 1
   3  =  3
   4  =  4
   5  =  4 + 1
   6  =  4 + 2
  17  =  11 + 5 + 1
  66  =  47 + 13 + 4 + 2
		

Crossrefs

Cf. A000032, A000045, A007895, A116543, A353655, A353657, A353658, A353659.

Programs

Mathematica

z = 120; 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[z]];
u = Take[Map[Last, t], z];
u1 = Map[Length, u]  (* A353655 *)
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[z]];
v = Take[Map[Last, t], z];
v1 = Map[Length, v]   (* A353656 *)
u1 - v1  (* A353657 *)
(* Peter J. C. Moses, May 04 2022 *)