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A256661 Rectangular array by antidiagonals: row n shows the numbers k such that R(k) consists of n terms, where R(k) is the minimal alternating Fibonacci representation of k.

%I A256661 #9 Jun 27 2017 15:43:02
%S A256661 1,2,4,3,6,9,5,7,14,25,8,10,15,38,64,13,11,17,40,98,169,21,12,22,41,
%T A256661 103,258,441,34,16,23,46,104,271,674,1156,55,18,24,59,106,273,708,
%U A256661 1766,3025,89,19,27,61,119,274,713,1855,4622,7921,144,20,28,62,153
%N A256661 Rectangular array by antidiagonals:  row n shows the numbers k such that R(k) consists of n terms, where R(k) is the minimal alternating Fibonacci representation of k.
%C A256661 See A256655 for definitions.  Every positive integer occurs exactly once.
%C A256661 (row 1):  A000045 (Fibonacci numbers)
%C A256661 (col 1):  A007598 (squared Fibonacci numbers)
%C A256661 (col 2):  A127546 (conjectured)
%e A256661 Northwest corner:
%e A256661 1     2     3     5     8     13    21
%e A256661 4     6     7     10    11    12    62
%e A256661 9     14    15    17    22    23    24
%e A256661 25    38    40    41    46    59    61
%e A256661 64    98    103   104   106   119   153
%e A256661 169   258   271   273   274   279   313
%e A256661 R(1) = 1, in row 1
%e A256661 R(2) = 2, in row 1
%e A256661 R(3) = 3, in row 1
%e A256661 R(4) = 5 - 1, in row 2
%e A256661 R(9) = 13 - 5 + 1, in row 3
%e A256661 R(25) = 34 - 13 + 5 - 1, in row 4
%e A256661 R(64) = 89 - 34 + 13 - 5 + 1, in row 5
%t A256661 b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
%t A256661 h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
%t A256661 g = h[23];  r[0] = {0};
%t A256661 r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
%t A256661 u = Table[Length[r[n]], {n, 1, 6000}];
%t A256661 TableForm[Table[Flatten[Position[u, k]], {k, 1, 9}]]
%Y A256661 Cf. A000045, A256655, A007598, A127546.
%K A256661 nonn,easy,tabl
%O A256661 1,2
%A A256661 _Clark Kimberling_, Apr 08 2015