A200714 Stolarsky representation interpreted as binary to decimal integers.

0, 1, 3, 2, 7, 5, 6, 15, 4, 11, 13, 14, 31, 10, 9, 23, 12, 27, 29, 30, 63, 8, 21, 19, 22, 47, 26, 25, 55, 28, 59, 61, 62, 127, 20, 17, 43, 18, 39, 45, 46, 95, 24, 53, 51, 54, 111, 58, 57, 119, 60, 123, 125, 126, 255, 16, 41, 35, 42, 87, 37, 38, 79, 44, 91, 93

Offset: 1

Author

Casey Mongoven, Nov 20 2011

Keywords

• nonn
• base

Comments

See explanation of Stolarsky representations in the C. Mongoven link.

Examples

			The Stolarsky representation of 19 is 11101. In binary this is equal to 29. So a(19) = 29.
		

Links

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

Crossrefs

Cf. A048679, A200648, A200649, A200650, A200651.

Cf. A000120, A005811, A023416, A070939.

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_] := FromDigits[stol[n], 2]; Array[a, 100] (* Amiram Eldar, Jul 07 2023 *)

PARI

a(n) = {if (n == 1, return (0)); tau = (1 + sqrt(5))/2; mn = 0; while ((m = round(mn*tau)) < n, mn++;); if (m == n, return (2*a(mn)+1)); mn = 0; while ((m = floor(mn*(1+tau)-tau/2)) < n, mn++;); if (m == n, return (2*a(mn))); error("neither A nor B !!");} \\ (cf C. Mongoven link) Michel Marcus, May 21 2013, Sep 02 2013

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) = fromdigits(stol(n), 2); \\ Amiram Eldar, Jul 07 2023

Formula

From Amiram Eldar, Jul 07 2023: (Start)

A000120(a(n)) = A200649(n).

A023416(a(n)) = A200650(n).

A070939(a(n)) = A200648(n).

A005811(a(n)) = A200651(n). (End)

Conjecture: a(n) = A367306(A358654(n-1)). - Mikhail Kurkov, Oct 17 2024

Extensions

More terms from Amiram Eldar, Jul 07 2023