A364127 The number of trailing 0's in the Stolarsky representation of n (A364121).

0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 3, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 3, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 4, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 3, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 1

Offset: 2

Author

Amiram Eldar, Jul 07 2023

Keywords

• nonn
• base

Comments

The first position of k = 2, 3, 4, ... is A055588(k+1).

The asymptotic density of the occurrences of k = 0, 1, 2, ... is (2-phi)^k/phi, where phi is the golden ratio (A001622).

The asymptotic mean of this sequence is phi - 1 (A094214) and the asymptotic standard deviation is 1.

Links

• Amiram Eldar, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A001622, A055588, A094214, A122840, A364121.

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_] := IntegerExponent[FromDigits[stol[n]], 10]; Array[a, 100, 2]

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) = valuation(fromdigits(stol(n)), 10);

Formula

a(n) = A122840(A364121(n)).