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A353656
Number of terms in the Lucas-Fibonacci representation of n.
5
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
1,2
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. (See Example.)
EXAMPLE
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
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 *)
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 04 2022
STATUS
approved