A036555 Hamming weight of 3n: number of 1's in binary expansion of 3n.

0, 2, 2, 2, 2, 4, 2, 3, 2, 4, 4, 2, 2, 4, 3, 4, 2, 4, 4, 4, 4, 6, 2, 3, 2, 4, 4, 3, 3, 5, 4, 5, 2, 4, 4, 4, 4, 6, 4, 5, 4, 6, 6, 2, 2, 4, 3, 4, 2, 4, 4, 4, 4, 6, 3, 4, 3, 5, 5, 4, 4, 6, 5, 6, 2, 4, 4, 4, 4, 6, 4, 5, 4, 6, 6, 4, 4, 6, 5, 6, 4, 6, 6, 6, 6, 8, 2, 3, 2, 4, 4, 3, 3, 5, 4, 5, 2, 4, 4

Offset: 0

Author

Dan Hoey

Keywords

• nonn
• base
• easy
• nice

Comments

a(n) is also the largest integer such that 2^a(n) divides binomial(6n,3n)=A066802(n). - Benoit Cloitre, Mar 27 2002

a(n) = A000120(A008585(n)). - Reinhard Zumkeller, Nov 03 2010

a(A002450(n)) = 2*n.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.
• Philippe Flajolet, Peter Grabner, Peter Kirschenhofer, Helmut Prodinger and Robert F. Tichy, Mellin transforms and asymptotics: digital sums, Theoret. Comput. Sci. 123 (1994), 291-314.
• Michael Gilleland, Some Self-Similar Integer Sequences
• D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721.

Crossrefs

Cf. A036556, A036557, A099033, A190944.

Programs

Haskell

a036555 = a000120 . (* 3)  -- Reinhard Zumkeller, Sep 01 2013

Maple

t1:=[];
for n from 0 to 100 do t2:=convert(3*n,base,2); t3:=add(t2[i],i=1..nops(t2)); t1:=[op(t1),t3];od:
t1;

Mathematica

Total/@IntegerDigits[3Range[0,100],2] (* Harvey P. Dale, Oct 03 2011 *)

PARI

a(n) = hammingweight(3*n); \\ Michel Marcus, Mar 13 2014

Extensions

Name edited by Michel Marcus, Mar 13 2014