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.)
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
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 02 2022
STATUS
approved