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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.
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%I #9 Jun 27 2017 15:43:02

%S 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 103,258,441,34,16,23,46,104,271,674,1156,55,18,24,59,106,273,708,

%U 1766,3025,89,19,27,61,119,274,713,1855,4622,7921,144,20,28,62,153

%N 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 See A256655 for definitions. Every positive integer occurs exactly once.

%C (row 1): A000045 (Fibonacci numbers)

%C (col 1): A007598 (squared Fibonacci numbers)

%C (col 2): A127546 (conjectured)

%e Northwest corner:

%e 1 2 3 5 8 13 21

%e 4 6 7 10 11 12 62

%e 9 14 15 17 22 23 24

%e 25 38 40 41 46 59 61

%e 64 98 103 104 106 119 153

%e 169 258 271 273 274 279 313

%e R(1) = 1, in row 1

%e R(2) = 2, in row 1

%e R(3) = 3, in row 1

%e R(4) = 5 - 1, in row 2

%e R(9) = 13 - 5 + 1, in row 3

%e R(25) = 34 - 13 + 5 - 1, in row 4

%e R(64) = 89 - 34 + 13 - 5 + 1, in row 5

%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[23]; r[0] = {0};

%t r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];

%t u = Table[Length[r[n]], {n, 1, 6000}];

%t TableForm[Table[Flatten[Position[u, k]], {k, 1, 9}]]

%Y Cf. A000045, A256655, A007598, A127546.

%K nonn,easy,tabl

%O 1,2

%A _Clark Kimberling_, Apr 08 2015