OFFSET
1,1
COMMENTS
For n>=0, row n is the monotonic sequence of positive integers m such that the number of even-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 odd-indexed Fibonacci numbers, see A165279.
EXAMPLE
Northwest corner:
2....5....7...13...15...18...20...34...36...
1....3....6....8...10...14...16...19...20...
4....9...11...17...22...24...27...29...31...
12..25...30...32...46...59...64...66...72...
Examples:
20=13+5+2=F(7)+F(5)+F(3), zero evens, so 20 is in row 0.
19=13+5+1=F(7)+F(5)+F(2), one even, so 19 is in row 1.
22=21+1=F(8)+F(2), two evens, so 22 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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Sep 13 2009
EXTENSIONS
More terms from Amiram Eldar, Feb 04 2020
STATUS
approved