A200649 Number of 1's in the Stolarsky representation of n.

0, 1, 2, 1, 3, 2, 2, 4, 1, 3, 3, 3, 5, 2, 2, 4, 2, 4, 4, 4, 6, 1, 3, 3, 3, 5, 3, 3, 5, 3, 5, 5, 5, 7, 2, 2, 4, 2, 4, 4, 4, 6, 2, 4, 4, 4, 6, 4, 4, 6, 4, 6, 6, 6, 8, 1, 3, 3, 3, 5, 3, 3, 5, 3, 5, 5, 5, 7, 3, 3, 5, 3, 5, 5, 5, 7, 3, 5, 5, 5, 7, 5, 5, 7, 5, 7, 7

Offset

1,3

Comments

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

Links

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

Casey Mongoven, Description of Stolarsky Representations.

Formula

a(n) = a(n - A130312(n-1)) + (A072649(n-1) - A072649(n - A130312(n-1) - 1)) mod 2 for n > 2 with a(1) = 0, a(2) = 1. - Mikhail Kurkov, Oct 19 2021 [verification needed]

a(n) = A200648(n) - A200650(n). - Amiram Eldar, Jul 07 2023

Example

The Stolarsky representation of 19 is 11101. This has 4 1's. So a(19) = 4.

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_] := Count[stol[n], 1]; Array[a, 100] (* Amiram Eldar, Jul 07 2023 *)

Prog

(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) = vecsum(stol(n)); \\ Amiram Eldar, Jul 07 2023

Crossrefs

Cf. A072649, A130312, A135818, A200648, A200650, A200651.

Sequence in context: A199246 A367858 A363718 * A298742 A086415 A069013

Adjacent sequences: A200646 A200647 A200648 * A200650 A200651 A200652

Keyword

nonn,base

Author

Casey Mongoven, Nov 19 2011

Extensions

More terms from Amiram Eldar, Jul 07 2023

Status

approved