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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).
2

%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