A050605 Column/row 2 of A050602: a(n) = add3c(n,2).

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

Offset: 0

Author

Antti Karttunen, Jun 22 1999

Keywords

• nonn

Comments

It seems that (n - Sum_{k=1..n} a(k) )/log(n) is bounded. - Benoit Cloitre, Oct 03 2002

2^a(n-1) is the highest power of 2 dividing the triangular number A000217(n) = n*(n+1)/2, for n >= 1. - Benoit Cloitre, Oct 03 2002 [corrected and rewritten by Wolfdieter Lang, Nov 21 2019]

a(n) is the number of trailing 0's in the binary reflected Gray code of n+1 (A014550). - Amiram Eldar, May 15 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Intrinsic Properties of a Non-Symmetric Number Triangle, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.

Crossrefs

Bisection gives column/row 1 of A050602: A007814.

Cf. A000217, A001511, A161737, A069834, A014550.

Programs

Magma

[Valuation(n*(n+1)/2, 2): n in [1..120]]; // Vincenzo Librandi, Aug 11 2017

Maple

with(Bits): add3c := proc(a, b) option remember; `if`(0 = And(a, b), 0, 1 + add3c(Xor(a, b), 2*And(a, b))) end: A050605 := n -> add3c(n, 2):
seq(A050605(n), n=0..80); # Johannes W. Meijer, Jun 18 2009; updated by Peter Luschny, Jul 12 2019

Mathematica

Table[IntegerExponent[(n + 1)(n + 2)/2, 2], {n, 0, 100}] (* Jean-François Alcover, Mar 04 2016 *)

PARI

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

Formula

a(4*n+2) = A001511(n). - Johannes W. Meijer, Jun 18 2009

a(n) = A007814(n+1) + A007814(n+2) - 1. - Ridouane Oudra, Oct 08 2019