OFFSET
1,2
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. Every positive integer occurs exactly once in the array.
EXAMPLE
Northwest corner:
1 2 3 5 8 13 21 34
4 6 9 11 12 14 16 17
7 10 15 18 19 23 26 27
49 51 70 72 77 79 82 88
80 83 114 117 125 128 133 143
549 551 570 572 782 784 803 805
889 892 923 926 1266 1269 1300 1303
6094 6096 6115 6117 6327 6329 6348 6350
MATHEMATICA
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[8000]];
Length[t];
u = Table[Length[t[[n]][[2]]], {n, 1, Length[t]}];
Take[u, 150]
TableForm[Table[Flatten[Position[u, k]], {k, 1, 8}]]
w[k_, n_] := Flatten[Position[u, k]][[n]]
Table[w[n - k + 1, k], {n, 8}, {k, n, 1, -1}] // Flatten
(* Peter J. C. Moses, May 04 2022 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, May 04 2022
STATUS
approved