A165279 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-indexed Fibonacci numbers in its Zeckendorf representation.

1, 3, 2, 4, 5, 7, 8, 6, 15, 20, 9, 10, 18, 41, 54, 11, 13, 19, 49, 109, 143, 12, 14, 28, 52, 130, 287, 376, 21, 16, 36, 53, 138, 342, 753, 986, 22, 17, 39, 75, 141, 363, 897, 1973, 2583, 24, 23, 40, 96, 142, 371, 952, 2350, 5167, 6764, 25, 26, 44, 104, 198, 374

Offset: 1

Author

Clark Kimberling, Sep 13 2009

Keywords

• nonn
• tabl

Comments

For n>=0, row n is the monotonic sequence of positive integers m such that the number of odd-indexed Fibonacci numbers in the Zeckendorf representation of m is n.

We begin the indexing at 2; that is, 1=F(2), 2=F(3), 3=F(4), 5=F(5),...

Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.

For counts of even-indexed Fibonacci numbers, see A165278.

Essentially, (row 0)=A054204, (column 1)=A035508.

Examples

			Northwest corner:
1....3....4....8....9...11...12...21...22...
2....5....6...10...13...14...16...17...23...
7...15...18...19...28...36...39...40...44...
20..41...49...52...53...75...96..104..107...
Examples:
12=8+3+1=F(6)+F(4)+F(2), zero odds, so 12 is in row 0.
28=21+5+2=F(8)+F(5)+F(3), two odds, so 28 is in row 2.
		

Crossrefs

Cf. A165276, A165277, A165278.

Programs

Mathematica

f[n_] := Module[{i = Ceiling[Log[GoldenRatio, Sqrt[5]*n]], v = {}, m = n}, While[i > 1, If[Fibonacci[i] <= m, AppendTo[v, 1]; m -= Fibonacci[i], If[v != {}, AppendTo[v, 0]]]; i--]; Total[Reverse[v][[1 ;; -1 ;; 2]]]]; T = GatherBy[SortBy[ Range[10^4], f], f]; Table[Table[T[[n - k + 1, k]], {k, n, 1, -1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020 *)

Extensions

More terms from Amiram Eldar, Feb 04 2020