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A353659
Rectangular array read by downwards antidiagonals: row k lists the numbers whose Lucas-Fibonacci representation has k terms.
3
1, 3, 2, 4, 5, 15, 7, 6, 17, 25, 11, 8, 22, 28, 172, 18, 9, 24, 36, 174, 279, 29, 10, 27, 39, 193, 282, 1913, 47, 12, 33, 44, 195, 313, 1915, 3096, 76, 13, 35, 54, 248, 316, 1934, 3099, 21221, 123, 14, 38, 57, 250, 402, 1936, 3130, 21223, 34337
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. Every positive integer occurs exactly once in this array.
EXAMPLE
Northwest corner:
1 3 4 7 11 18 29 47 76 123
2 5 6 8 9 10 12 13 14 16
15 17 22 24 27 33 35 38 40 41
25 28 36 39 44 54 57 62 65 66
172 174 193 195 248 250 269 271 276 278
279 282 313 316 402 405 436 439 447 450
1913 1915 1934 1936 2146 2148 2167 2169 2756 2758
3096 3099 3130 3133 3473 3476 3507 3510 4460 4463
MATHEMATICA
fib = Map[Fibonacci, Range[2, 51]];
luc = Map[LucasL, Range[1, 50]];
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[50000]];
Length[t];
u = Table[Length[t[[n]][[2]]], {n, 1, Length[t]}];
Take[u, 150]
TableForm[Table[Flatten[Position[u, k]], {k, 1, 11}]];
w[k_, n_] := Flatten[Position[u, k]][[n]]
Table[w[n - k + 1, k], {n, 11}, {k, n, 1, -1}] // Flatten
(* Peter J. C. Moses, May 04 2022 *)
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, May 04 2022
STATUS
approved