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

1,3

Links

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

Casey Mongoven, Description of Stolarsky Representations.

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)

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]

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) = fromdigits(stol(n));

Crossrefs

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

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

Sequence in context: A367547 A070836 A063432 * A038324 A164125 A212529

Adjacent sequences: A364118 A364119 A364120 * A364122 A364123 A364124

Keyword

nonn,base

Author

Amiram Eldar, Jul 07 2023

Status

approved