A275019 2-adic valuation of tetrahedral numbers C(n+2,3) = n(n+1)(n+2)/6 = A000292(n).

0, 2, 1, 2, 0, 3, 2, 3, 0, 2, 1, 2, 0, 4, 3, 4, 0, 2, 1, 2, 0, 3, 2, 3, 0, 2, 1, 2, 0, 5, 4, 5, 0, 2, 1, 2, 0, 3, 2, 3, 0, 2, 1, 2, 0, 4, 3, 4, 0, 2, 1, 2, 0, 3, 2, 3, 0, 2, 1, 2, 0, 6, 5, 6, 0, 2, 1, 2, 0, 3, 2, 3, 0, 2, 1, 2, 0, 4, 3, 4, 0, 2, 1, 2, 0, 3, 2, 3, 0, 2, 1, 2, 0, 5, 4, 5, 0, 2, 1, 2, 0, 3, 2, 3, 0, 2, 1, 2, 0, 4

Offset: 1

Author

M. F. Hasler, Dec 03 2016

Keywords

• nonn

Comments

The subsequence of every other term (a(2n-1), n >= 1) is the ruler sequence A007814 = (0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, ...), in particular every fourth term is zero. The nonzero terms among them, a(4n-1) = A007814(2n) (n >= 1) have both their neighbors equal to one more than themselves, a(4n-2) = a(4n) = a(4n-1) + 1 = A007814(2n) + 1.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000292, A007814.

Programs

Magma

[Valuation(n*(n+1)*(n+2)/6, 2): n in [1..100]]; // Vincenzo Librandi, Dec 04 2016

Maple

seq(padic:-ordp(n*(n+1)*(n+2)/6,2),n=1..100); # Robert Israel, Dec 04 2016

Mathematica

a[n_] := IntegerExponent[n*(n+1)*(n+2)/6, 2]; Array[a, 100] (* Amiram Eldar, Sep 13 2024 *)

PARI

a(n)=valuation(n*(n+1)*(n+2)/6,2)

Python

def A275019(n): return (~(m:=n*(n+1)*(n+2)//6)& m-1).bit_length() # Chai Wah Wu, Jul 07 2022

Formula

From Robert Israel, Dec 04 2016: (Start)

a(n) = A007814(n) + A007814(n+1) + A007814(n+2) - 1.

G.f.: (1+x+x^2)*Sum_{k>=1} x^(2^k-2)/(1-x^(2^k)) - 1/(1-x). (End)

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Sep 13 2024