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A256659 Rectangular array by antidiagonals: row n consists of numbers k such that -F(n+1) is the trace of the minimal alternating Fibonacci representation of k, where F = A000045 (Fibonacci numbers).

%I A256659 #5 Apr 09 2015 07:59:14
%S A256659 4,7,6,12,11,10,20,19,18,16,25,32,31,29,26,33,40,52,50,47,42,38,53,65,
%T A256659 84,81,76,68,41,61,86,105,136,131,123,110,46,66,99,139,170,220,212,
%U A256659 199,178,54,74,107,160,225,275,356,343,322,288,59,87,120,173,259
%N A256659 Rectangular array by antidiagonals:  row n consists of numbers k such that -F(n+1) is the trace of the minimal alternating Fibonacci representation of k, where F = A000045 (Fibonacci numbers).
%C A256659 See A256655 for definitions.  This array and the array at A256658 partition the positive integers.  The row differences are Fibonacci numbers.   The columns satisfy the Fibonacci recurrence x(n) = x(n-1) + x(n-2).
%e A256659 Northwest corner:
%e A256659 4    7    12    20    25    33    38    41    46
%e A256659 6    11   19    32    40    53    61    66    74
%e A256659 10   18   31    52    65    86    99    102   120
%e A256659 16   29   50    84    105   139   160   173   194
%e A256659 26   47   81    136   170   225   259   280   314
%t A256659 b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
%t A256659 h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
%t A256659 g = h[18];  r[0] = {0};
%t A256659 r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
%t A256659 t = Table[Last[r[n]], {n, 0, 1000}];  (* A256656 *)
%t A256659 TableForm[Table[Flatten[-1 + Position[t, b[n]]], {n, 2, 8}]]   (* A256658 *)
%t A256659 TableForm[Table[Flatten[-1 + Position[t, -b[n]]], {n, 2, 8}]]  (* A256659 *)
%Y A256659 Cf. A000045, A256655, A246658.
%K A256659 nonn,tabl,easy
%O A256659 1,1
%A A256659 _Clark Kimberling_, Apr 08 2015