A200648 Length of Stolarsky representation of n.

1, 1, 2, 2, 3, 3, 3, 4, 3, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 5, 6, 4, 5, 5, 5, 6, 5, 5, 6, 5, 6, 6, 6, 7, 5, 5, 6, 5, 6, 6, 6, 7, 5, 6, 6, 6, 7, 6, 6, 7, 6, 7, 7, 7, 8, 5, 6, 6, 6, 7, 6, 6, 7, 6, 7, 7, 7, 8, 6, 6, 7, 6, 7, 7, 7, 8, 6, 7, 7, 7, 8, 7, 7, 8, 7, 8, 8

Offset: 1

Author

Casey Mongoven, Nov 19 2011

Keywords

• nonn
• base

Comments

For the Stolarsky representation of n, see the C. Mongoven link.

Conjecture: a(n) is the sum of row n-1 of A385886. To obtain it, first take maximal anti-run lengths of binary indices of each nonnegative integer (giving A384877), then remove all duplicate rows (giving A385886), and finally take the sum of each remaining row. For length instead of sum we appear to have A200649. - Gus Wiseman, Jul 21 2025

Examples

			The Stolarsky representation of 19 is 11101. This is of length 5. So a(19) = 5.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Casey Mongoven, Description of Stolarsky Representations.

Crossrefs

Cf. A135817, A200651.

Counting just ones gives A200649.

Counting just zeros gives A200650.

Stolarsky representation is listed by A385888, ranks A200714.

A000120 counts 1's in binary expansion.

A384890 counts maximal anti-runs of binary indices, ranks A385816.

A385886 lists maximal anti-run lengths of binary indices.

Cf. A023758, A048793, A052499, A069010, A072649, A083368, A135818, A245562, A245563, A348366, A384877, A384893.

Programs

Mathematica

stol[n_] := stol[n] = If[n == 1, {}, If[n != Round[Round[n/GoldenRatio]*GoldenRatio], Join[stol[Floor[n/GoldenRatio^2] + 1], {0}], Join[stol[Round[n/GoldenRatio]], {1}]]];
a[n_] := If[n == 1, 1, Length[stol[n]]]; Array[a, 100] (* Amiram Eldar, Jul 07 2023 *)

PARI

stol(n) = {my(phi=quadgen(5)); if(n==1, [], if(n != round(round(n/phi)*phi), concat(stol(floor(n/phi^2) + 1), [0]), concat(stol(round(n/phi)), [1])));}
a(n) = if(n == 1, 1, #stol(n)); \\ Amiram Eldar, Jul 07 2023

Formula

a(n) = A200649(n) + A200650(n). - Michel Marcus, Mar 14 2023

Extensions

More terms from Amiram Eldar, Jul 07 2023