A353655 Number of terms in the Fibonacci-Lucas representation of n.

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 2, 2, 1, 2, 3, 2, 2, 3, 3, 2, 1, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 1, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 4, 3, 4, 2, 3, 3, 1, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 4, 3, 4, 2, 3, 3, 3, 4, 3, 4, 5, 3, 4, 5, 2, 3, 3

Offset: 1

Author

Clark Kimberling, May 02 2022

Keywords

• nonn

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. (See Example.)

Examples

			  n      FL(n)
  1   =  1
  2   =  2
  3   =  3
  4   =  3 + 1
  5   =  5
  6   =  5 + 1
  33  =  21 + 11 + 1
  47  =  34 + 11 + 2
  83  =  55 + 18 + 8 + 1 + 1
		

Crossrefs

Cf. A000032, A000045, A007895, A116543, A353656, A353657.

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 *)