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

1,2

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.

Links

Table of n, a(n) for n=1..61.

Example

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.

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 *)

Crossrefs

Cf. A165276, A165277, A165278.

Sequence in context: A163241 A234027 A364839 * A345866 A125060 A039882

Adjacent sequences: A165276 A165277 A165278 * A165280 A165281 A165282

Keyword

nonn,tabl

Author

Clark Kimberling, Sep 13 2009

Extensions

More terms from Amiram Eldar, Feb 04 2020

Status

approved