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A006047
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Number of entries in n-th row of Pascal's triangle not divisible by 3.
(Formerly M0422)
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9
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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
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0,2
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COMMENTS
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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 n-th row of Pascal's triangle not divisible by a prime p, which is given by a simple recursion using \otimes, the Kronecker (or tensor) product of vectors. Let v_0=(1,2,...,p). Then v_{n+1)=v_0 \otimes v_n. The vector v_n contains the values for the first p^n rows of Pascal's triangle (rows 0 through p^n-1). - William B. Everett (bill(AT)chgnet.ru), Mar 29 2008
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REFERENCES
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J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=0..79.
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
Michael Gilleland, Some Self-Similar Integer Sequences
Sam Northshield, Sums across Pascal’s triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.
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FORMULA
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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). [From Johannes W. Meijer, Jun 5 2011]
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EXAMPLE
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15 in base 3 is 120, here r=1 and s=1 so a(15) = 3*2 = 6.
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MAPLE
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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); (from Emeric Deutsch)
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MATHEMATICA
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Nest[Flatten[ # /. a_Integer -> {a, 2a, 3a}] &, {1}, 4] (from Robert G. Wilson v, Jan 24 2006)
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PROG
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(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)))) (from Ralf Stephan)
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CROSSREFS
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Sequence in context: A215190 A214943 A202864 * A062068 A130542 A128502
Adjacent sequences: A006044 A006045 A006046 * A006048 A006049 A006050
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KEYWORD
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nonn
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AUTHOR
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Jeffrey Shallit
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EXTENSIONS
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More terms from Ralf Stephan, Sep 15 2003
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STATUS
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approved
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