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A353659 Rectangular array read by downwards antidiagonals: row k lists the numbers whose Lucas-Fibonacci representation has k terms.

%I A353659 #17 Jan 21 2023 03:01:00
%S A353659 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,
%T A353659 193,282,1913,47,12,33,44,195,313,1915,3096,76,13,35,54,248,316,1934,
%U A353659 3099,21221,123,14,38,57,250,402,1936,3130,21223,34337
%N A353659 Rectangular array read by downwards antidiagonals: row k lists the numbers whose Lucas-Fibonacci representation has k terms.
%C A353659 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.
%e A353659 Northwest corner:
%e A353659      1     3     4     7    11    18    29    47    76   123
%e A353659      2     5     6     8     9    10    12    13    14    16
%e A353659     15    17    22    24    27    33    35    38    40    41
%e A353659     25    28    36    39    44    54    57    62    65    66
%e A353659    172   174   193   195   248   250   269   271   276   278
%e A353659    279   282   313   316   402   405   436   439   447   450
%e A353659   1913  1915  1934  1936  2146  2148  2167  2169  2756  2758
%e A353659   3096  3099  3130  3133  3473  3476  3507  3510  4460  4463
%t A353659 fib = Map[Fibonacci, Range[2, 51]];
%t A353659 luc = Map[LucasL, Range[1, 50]];
%t A353659 t = Map[(n = #; lf = {}; f = 0; l = 0;
%t A353659      While[IntegerQ[f], n = n - l - f;
%t A353659       l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n &] - 1]];
%t A353659       f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n - l &] - 1]];
%t A353659       AppendTo[lf, {l, f}]];
%t A353659      {Total[#], #} &[Select[Flatten[lf], IntegerQ]]) &, Range[50000]];
%t A353659 Length[t];
%t A353659 u = Table[Length[t[[n]][[2]]], {n, 1, Length[t]}];
%t A353659 Take[u, 150]
%t A353659 TableForm[Table[Flatten[Position[u, k]], {k, 1, 11}]];
%t A353659 w[k_, n_] := Flatten[Position[u, k]][[n]]
%t A353659 Table[w[n - k + 1, k], {n, 11}, {k, n, 1, -1}] // Flatten
%t A353659 (* _Peter J. C. Moses_, May 04 2022 *)
%Y A353659 Cf. A000032, A000045, A007895, A116543, A353655, A353656, A353658.
%K A353659 nonn,tabl
%O A353659 1,2
%A A353659 _Clark Kimberling_, May 04 2022