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.

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, 103, 258, 441, 34, 16, 23, 46, 104, 271, 674, 1156, 55, 18, 24, 59, 106, 273, 708, 1766, 3025, 89, 19, 27, 61, 119, 274, 713, 1855, 4622, 7921, 144, 20, 28, 62, 153

Offset: 1

Author

Clark Kimberling, Apr 08 2015

Keywords

• nonn
• easy
• tabl

Comments

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

(row 1): A000045 (Fibonacci numbers)

(col 1): A007598 (squared Fibonacci numbers)

(col 2): A127546 (conjectured)

Examples

			Northwest corner:
1     2     3     5     8     13    21
4     6     7     10    11    12    62
9     14    15    17    22    23    24
25    38    40    41    46    59    61
64    98    103   104   106   119   153
169   258   271   273   274   279   313
R(1) = 1, in row 1
R(2) = 2, in row 1
R(3) = 3, in row 1
R(4) = 5 - 1, in row 2
R(9) = 13 - 5 + 1, in row 3
R(25) = 34 - 13 + 5 - 1, in row 4
R(64) = 89 - 34 + 13 - 5 + 1, in row 5
		

Crossrefs

Cf. A000045, A256655, A007598, A127546.

Programs

Mathematica

b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
g = h[23];  r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
u = Table[Length[r[n]], {n, 1, 6000}];
TableForm[Table[Flatten[Position[u, k]], {k, 1, 9}]]