

A165278


Table read by antidiagonals: T(n, k) is the kth number with n1 evenindexed Fibonacci numbers in its Zeckendorf representation.


5



2, 5, 1, 7, 3, 4, 13, 6, 9, 12, 15, 8, 11, 25, 33, 18, 10, 17, 30, 67, 88, 20, 14, 22, 32, 80, 177, 232, 34, 16, 24, 46, 85, 211, 465, 609, 36, 19, 27, 59, 87, 224, 554, 1219, 1596, 39, 21, 29, 64, 122, 229, 588, 1452, 3193, 4180, 41, 23, 31, 66, 156, 231, 601
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OFFSET

1,1


COMMENTS

For n>=0, row n is the monotonic sequence of positive integers m such that the number of evenindexed 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 oddindexed Fibonacci numbers, see A165279.
Essentially, (row 0)=A062879, (column 1)=A027941, (column 2)=A069403.


LINKS

Table of n, a(n) for n=1..62.


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

Cf. A165276, A165277, A165279.
Sequence in context: A325137 A321966 A304822 * A334379 A106619 A173630
Adjacent sequences: A165275 A165276 A165277 * A165279 A165280 A165281


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Sep 13 2009


EXTENSIONS

More terms from Amiram Eldar, Feb 04 2020


STATUS

approved



