A036557 -id:A036557 - cp's OEIS frontend

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

Dan Hoey

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.

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.

Cf. A036556, A036557, A099033, A190944.

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

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;

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

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

Name edited by Michel Marcus, Mar 13 2014

A036556 Integers which when multiplied by 3 have an odd number of 1's in their binary expansion (cf. A000069).

Original entry on oeis.org

7, 14, 23, 27, 28, 29, 31, 39, 46, 54, 56, 57, 58, 62, 71, 78, 87, 91, 92, 93, 95, 103, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 119, 123, 124, 125, 127, 135, 142, 151, 155, 156, 157, 159, 167, 174, 182, 184, 185, 186, 190, 199, 206, 214

Offset: 1

Dan Hoey

In other words, numbers n such that 3n is odious.
Numbers n such that valuation(binomial(3n,n),2) is odd. - Benoit Cloitre, Jun 06 2004
Intersection of A000069 and A008585 (multiples of 3), divided by 3.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000069, A036555, A036557, A180963.

[ n : n in [0..150] | IsOdd(&+Intseq(3*n, 2))]; // Vincenzo Librandi, Apr 13 2011

Select[ Range[ 214 ], OddQ[ Plus@@IntegerDigits[ 3#, 2 ]]& ]

for(n=1,214,if(valuation(binomial(3*n,n),2)%2==1,print1(n,","))) \\ Benoit Cloitre, Jun 06 2004

a(n) = A180963(n)/3. - Amiram Eldar, Aug 06 2023

Definition corrected by N. J. A. Sloane, Jan 09 2007

cp's OEIS Frontend

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

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