A364121 Stolarsky representation of n.

0, 1, 11, 10, 111, 101, 110, 1111, 100, 1011, 1101, 1110, 11111, 1010, 1001, 10111, 1100, 11011, 11101, 11110, 111111, 1000, 10101, 10011, 10110, 101111, 11010, 11001, 110111, 11100, 111011, 111101, 111110, 1111111, 10100, 10001, 101011, 10010, 100111, 101101

Offset: 1

Author

Amiram Eldar, Jul 07 2023

Keywords

• nonn
• base

Links

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

Crossrefs

Cf. A001622, A007064, A200648, A200649, A200650, A200651, A200714.

Cf. A007088, A043562, A055641, A055642, A268643.

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]]; Array[a, 100]

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));

Formula

Description of an algorithm for calculating a(n):

Let s(1) = {} be the empty set, and for n > 1, let s(n) be the sequence of digits of a(n). s(n) can be calculated recursively by:

1. If n = round(round(n/phi)*phi) then s(n) = s(floor(n/phi^2) + 1) U {0}, where phi is the golden ratio (A001622) and U denotes concatenation.

2. If n != round(round(n/phi)*phi) then s(n) = s(round(n/phi)) U {1}.

a(n) = A007088(A200714(n)).

A268643(a(n)) = A200649(n).

A055641(a(n)) = A200650(n).

A055642(a(n)) = A200648(n).

A043562(a(n)) = A200651(n)