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

1,4

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.)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

  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

Maple

Fibs:= [seq(combinat:-fibonacci(i), i=2..30)]:
Lucs:= [1, 3, seq(Fibs[n+1]+Fibs[n-1], n=2..28)]:
f:= proc(n) local x, i; uses ListTools;
  x:= n;
  for i from 1 do
    if i::odd then x:= x - Fibs[BinaryPlace(Fibs, x+1)]
    else x:= x - Lucs[BinaryPlace(Lucs, x+1)]
    fi;
    if x = 0 then return i fi;
  od
end proc:
map(f, [$1..100]); # Robert Israel, Jun 12 2026

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 *)

Crossrefs

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

Sequence in context: A057945 A374438 A285730 * A280055 A253092 A194546

Adjacent sequences: A353652 A353653 A353654 * A353656 A353657 A353658

Keyword

nonn,changed

Author

Clark Kimberling, May 02 2022

Status

approved