A200647 Number of equal bit-runs in Wythoff representation of n.

1, 1, 2, 2, 1, 2, 3, 2, 2, 3, 4, 2, 1, 2, 3, 4, 4, 3, 2, 3, 2, 2, 3, 4, 4, 3, 4, 5, 4, 2, 3, 4, 2, 1, 2, 3, 4, 4, 3, 4, 5, 4, 4, 5, 6, 4, 3, 2, 3, 4, 4, 3, 2, 3, 2, 2, 3, 4, 4, 3, 4, 5, 4, 4, 5, 6, 4, 3, 4, 5, 6, 6, 5, 4, 5, 4, 2, 3, 4, 4, 3, 4, 5, 4, 2, 3, 4

Offset

1,3

References

Wolfdieter Lang, The Wythoff and the Zeckendorf representations of numbers are equivalent, in G. E. Bergum et al. (edts.) Application of Fibonacci numbers vol. 6, Kluwer, Dordrecht, 1996, pp. 319-337. [See A317208 for a link.]

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Aviezri S. Fraenkel, From Enmity to Amity, American Mathematical Monthly, Vol. 117, No. 7 (2010) 646-648.

Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.

Example

The Wythoff representation of 29 is '10110'. This has 4 equal bit-runs: '1', '0', '11' and '0'. So a(29) = 4.

Mathematica

z[n_] := Floor[(n + 1)*GoldenRatio] - n - 1; h[n_] := z[n] - z[n - 1]; w[n_] := Module[{m = n, zm = 0, hm, s = {}}, While[zm != 1, hm = h[m]; AppendTo[s, hm]; If[hm == 1, zm = z[m], zm = z[z[m]]]; m = zm]; s]; a[n_] := Length[Split[w[n]]]; Array[a, 100] (* Amiram Eldar, Jul 01 2023 *)

Crossrefs

Cf. A135817, A317208.

Sequence in context: A090970 A091972 A025833 * A261625 A237284 A294186

Adjacent sequences: A200644 A200645 A200646 * A200648 A200649 A200650

Keyword

nonn

Author

Casey Mongoven, Nov 19 2011

Extensions

More terms from Amiram Eldar, Jul 01 2023

Status

approved