A352427 a(n) is the number of trailing 0's in the minimal representation of n in terms of the positive Pell numbers (A317204).

0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 4, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1, 4, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 5, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 3, 0, 1, 0, 1

Offset: 1

Author

Amiram Eldar, Mar 16 2022

Keywords

• nonn
• base

Comments

The asymptotic density of the occurrences of 0 is sqrt(2)-1 and of the occurrences of k = 1, 2, ... is 2*(sqrt(2)-1)^(k+1).

The asymptotic mean of this sequence is 1 and its asymptotic variance is sqrt(2).

Links

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

Crossrefs

Cf. A000129, A003151, A003152, A286666, A286667, A317204.

Similar sequences: A003849 (dual Zeckendorf), A035614 (Zeckendorf), A230403 (factorial), A276084 (primorial), A278045 (tribonacci).

Programs

Mathematica

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; a[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerExponent[Total[3^(s - 1)], 3]]; Array[a, 100]

Formula

a(A000129(n)) = n-1 for n>=1.

a(n) = 0 if and only if n is in A286666.

a(n) > 0 if and only if n is in A286667.

a(n) == 0 (mod 2) if and only if n is in A003152.

a(n) == 1 (mod 2) if and only if n is in A003151.