A358360 - OEIS

A358360

The 3-adic valuation of the central Delannoy numbers (sequence A001850).

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

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OFFSET

0,4

COMMENTS

The 3-adic valuation of x is the exponent of the highest power of 3 dividing x.

REFERENCES

J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press (2003), p. 453.

LINKS

Table of n, a(n) for n=0..86.

Max A. Alekseyev, Tewodros Amdeberhan, Jeffrey Shallit, and Ingrid Vukusic, On the p-adic valuations of values of Legendre polynomials, Res. Num. Theory 12 (2026), Art. 52. See also arXiv:2505.08935 [math.NT], 2025. See p. 3.

Jeffrey Shallit, k-regular sequences. Talk for the Colloque Michel Mendès France, Bordeaux, France, December 12 2000.

Zhao Shen, On a conjecture of J. Shallit about Apéry-like numbers, arXiv:2112.11135 [math.NT], 2021.

FORMULA

b(n) = b(floor(n/3)) + (floor(n/3) mod 2), if n == 0,2 (mod 3);

b(n) = b(floor(n/9)) + 1, if n == 1 (mod 3).

a(n) = A007949(A001850(n)).

Conjectures from Ridouane Oudra, May 20 2026: (Start)

a(n) = Sum_{k=1..n} (-1)^(k+1)*A051064(k).

a(n) = Sum_{k>=0} (floor(n/3^k) mod 2).

a(n) = A000989(floor((n+1)/2)) + A168570(n+1).

a(2*n) = A000989(n).

a(2*n+1) = A000989(n) + A254046(n+1). (End)