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A353658 Rectangular array by antidiagonals: row k lists the numbers whose Fibonacci-Lucas representation has k terms.

%I A353658 #22 Dec 26 2024 23:35:42
%S A353658 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,
%T A353658 114,551,889,34,16,23,77,117,570,892,6094,55,17,26,79,125,572,923,
%U A353658 6096,9861,89,20,27,82,128,782,926,6115,9864,67589
%N A353658 Rectangular array by antidiagonals: row k lists the numbers whose Fibonacci-Lucas representation has k terms.
%C A353658 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.
%e A353658 Northwest corner:
%e A353658      1     2     3     5     8    13    21    34
%e A353658      4     6     9    11    12    14    16    17
%e A353658      7    10    15    18    19    23    26    27
%e A353658     49    51    70    72    77    79    82    88
%e A353658     80    83   114   117   125   128   133   143
%e A353658    549   551   570   572   782   784   803   805
%e A353658    889   892   923   926  1266  1269  1300  1303
%e A353658   6094  6096  6115  6117  6327  6329  6348  6350
%t A353658 fib = Map[Fibonacci, Range[2, 51]];
%t A353658 luc = Map[LucasL, Range[1, 50]];
%t A353658 t = Map[(n = #; fl = {}; f = 0; l = 0;
%t A353658      While[IntegerQ[l], n = n - f - l;
%t A353658       f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n &] - 1]];
%t A353658       l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n - f &] - 1]];
%t A353658       AppendTo[fl, {f, l}]];
%t A353658      {Total[#], #} &[Select[Flatten[fl], IntegerQ]]) &, Range[8000]];
%t A353658 Length[t];
%t A353658 u = Table[Length[t[[n]][[2]]], {n, 1, Length[t]}];
%t A353658 Take[u, 150]
%t A353658 TableForm[Table[Flatten[Position[u, k]], {k, 1, 8}]]
%t A353658 w[k_, n_] := Flatten[Position[u, k]][[n]]
%t A353658 Table[w[n - k + 1, k], {n, 8}, {k, n, 1, -1}] // Flatten
%t A353658 (* _Peter J. C. Moses_, May 04 2022 *)
%Y A353658 Cf. A000032, A000045, A007895, A116543, A353655, A353656, A353659.
%K A353658 nonn,tabl
%O A353658 1,2
%A A353658 _Clark Kimberling_, May 04 2022