%I #43 Sep 13 2024 06:54:43

%S 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,

%T 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,

%U 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

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

%C 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.

%H Robert Israel, <a href="/A275019/b275019.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Robert Israel_, Dec 04 2016: (Start)

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

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

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

%p seq(padic:-ordp(n*(n+1)*(n+2)/6,2),n=1..100); # _Robert Israel_, Dec 04 2016

%t a[n_] := IntegerExponent[n*(n+1)*(n+2)/6, 2]; Array[a, 100] (* _Amiram Eldar_, Sep 13 2024 *)

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

%o (Magma) [Valuation(n*(n+1)*(n+2)/6, 2): n in [1..100]]; // _Vincenzo Librandi_, Dec 04 2016

%o (Python)

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

%Y Cf. A000292, A007814.

%K nonn

%O 1,2

%A _M. F. Hasler_, Dec 03 2016