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%I #4 Apr 09 2015 07:59:06
%S 1,9,2,14,15,3,17,23,24,5,22,28,37,39,8,27,36,45,60,63,13,30,44,58,73,
%T 97,102,21,35,49,71,94,118,157,165,34,43,57,79,115,152,191,254,267,55,
%U 48,70,92,128,186,246,309,411,432,89,51,78,113,149,207,301
%N 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 See A256655 for definitions. This array and the array at A256659 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 Northwest corner:
%e 1 9 14 17 22 27 30 35 43
%e 2 15 23 28 36 44 49 57 70
%e 3 24 37 45 58 71 79 92 113
%e 5 39 69 73 94 115 128 149 183
%e 8 63 97 118 152 186 207 241 296
%e 13 102 157 191 246 301 335 390 479
%t b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
%t h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
%t g = h[18]; r[0] = {0};
%t r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
%t t = Table[Last[r[n]], {n, 0, 1000}]; (* A256656 *)
%t TableForm[Table[Flatten[-1 + Position[t, b[n]]], {n, 2, 8}]] (* A256658 *)
%t TableForm[Table[Flatten[-1 + Position[t, -b[n]]], {n, 2, 8}]] (* A256659 *)
%Y Cf. A000045, A256655, A256659.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Apr 08 2015