|
|
A353658
|
|
Rectangular array by antidiagonals: row k lists the numbers whose Fibonacci-Lucas representation has k terms.
|
|
3
|
|
|
1, 2, 4, 3, 6, 7, 5, 9, 10, 49, 8, 11, 15, 51, 80, 13, 12, 18, 70, 83, 549, 21, 14, 19, 72, 114, 551, 889, 34, 16, 23, 77, 117, 570, 892, 6094, 55, 17, 26, 79, 125, 572, 923, 6096, 9861, 89, 20, 27, 82, 128, 782, 926, 6115, 9864, 67589
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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.
|
|
LINKS
|
|
|
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
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|