

A006047


Number of entries in nth row of Pascal's triangle not divisible by 3.
(Formerly M0422)


14



1, 2, 3, 2, 4, 6, 3, 6, 9, 2, 4, 6, 4, 8, 12, 6, 12, 18, 3, 6, 9, 6, 12, 18, 9, 18, 27, 2, 4, 6, 4, 8, 12, 6, 12, 18, 4, 8, 12, 8, 16, 24, 12, 24, 36, 6, 12, 18, 12, 24, 36, 18, 36, 54, 3, 6, 9, 6, 12, 18, 9, 18, 27, 6, 12, 18, 12, 24, 36, 18, 36, 54, 9, 18, 27, 18, 36, 54, 27, 54
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OFFSET

0,2


COMMENTS

Fixed point of the morphism a > a, 2a, 3a, starting from a(1) = 1.  Robert G. Wilson v, Jan 24 2006
This is a particular case of the number of entries in nth row of Pascal's triangle not divisible by a prime p, which is given by a simple recursion using ⊗, the Kronecker (or tensor) product of vectors. Let v_0=(1,2,...,p). Then v_{n+1)=v_0 ⊗ v_n, where the vector v_n contains the values for the first p^n rows of Pascal's triangle (rows 0 through p^n1).  William B. Everett (bill(AT)chgnet.ru), Mar 29 2008
a(n) = A206424(n) + A227428(n); number of nonzero terms in row n of triangle A083093.  Reinhard Zumkeller, Jul 11 2013


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
J.P. Allouche and J. Shallit, The ring of kregular sequences, Theoretical Computer Sci., 98 (1992), 163197.
Michael Gilleland, Some SelfSimilar Integer Sequences
Sam Northshield, Sums across Pascal’s triangle modulo 2, Congressus Numerantium, 200, pp. 3552, 2010.


FORMULA

Write n in base 3; if the representation contains r 1's and s 2's then a(n) = 3^s * 2^r. Also a(n) = Sum_{k=0, 1, .., n} (C(n, k)^2 mod 3).  Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
a(n) = b(n+1), with b(1)=1, b(2)=2, b(3n)=3b(n), b(3n+1)=b(n+1), b(3n+2)=2b(n+1).  Ralf Stephan, Sep 15 2003
G.f.: prod(n>=0, (1+2*x^(3^n)+3*x^(2*3^n))) (Northshield).  Johannes W. Meijer, Jun 05 2011


EXAMPLE

15 in base 3 is 120, here r=1 and s=1 so a(15) = 3*2 = 6.
William B. Everett's comment with p=3, n=2: v_0 = (1,2,3), v_1 = (1,2,3) => v_2 = (1*1,1*2,1*3,2*1,2*2,2*3,3*1,3*2,3*3) = (1,2,3,2,4,6,3,6,9), the first 3^2 values of the present sequence.  Wolfdieter Lang, Mar 19 2014


MAPLE

p:=proc(n) local ct, k: ct:=0: for k from 0 to n do if binomial(n, k) mod 3 = 0 then else ct:=ct+1 fi od: end: seq(p(n), n=0..82); # Emeric Deutsch


MATHEMATICA

Nest[Flatten[ # /. a_Integer > {a, 2a, 3a}] &, {1}, 4] (* Robert G. Wilson v, Jan 24 2006 *)
Nest[ Join[#, 2#, 3#] &, {1}, 4] (* Robert G. Wilson v, Jul 27 2014 *)


PROG

(PARI) b(n)=if(n<3, n, if(n%3==0, 3*b(n/3), if(n%3==1, 1*b((n+2)/3), 2*b((n+1)/3)))) \\ Ralf Stephan
(Haskell)
a006047 = sum . map signum . a083093_row
 Reinhard Zumkeller, Jul 11 2013


CROSSREFS

Cf. A001316, A089898.
Sequence in context: A215190 A214943 A202864 * A062068 A130542 A128502
Adjacent sequences: A006044 A006045 A006046 * A006048 A006049 A006050


KEYWORD

nonn


AUTHOR

Jeffrey Shallit


EXTENSIONS

More terms from Ralf Stephan, Sep 15 2003


STATUS

approved



