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A081611
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Number of numbers <= n having no 2 in their ternary representation.
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6
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1, 2, 2, 3, 4, 4, 4, 4, 4, 5, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 12, 12, 13, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16
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OFFSET
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0,2
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COMMENTS
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a(n) is also the size of the subset of [1..n] when numbers are added greedily so as to not contain a 3-term arithmetic progression, i.e., according to A003278: a(n) = the largest k such that A003278(k) <= n. (Cf. A003002, the size of the optimal (largest) 3-free subset of [1..n].) - Shreevatsa R, Oct 19 2009
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LINKS
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FORMULA
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G.f. A(x) satisfies A(x)=(1+x)*(1+x+x^2)*A(x^3).
G.f.: (1/(1-x))*Product_{k>=0} (1+x^(3^k)).
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MATHEMATICA
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CoefficientList[Series[Product[1+x^(3^k), {k, 0, 5}]/(1-x), {x, 0, 100}], x] (* G. C. Greubel, Mar 29 2019 *)
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PROG
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(PARI) {a(n)=local(A, m); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=3; A=(1+x)*(1+x+x^2)*subst(A, x, x^3)); polcoeff(A, n))} /* Michael Somos, Aug 31 2006 */
(PARI) my(x='x+O('x^100)); Vec(prod(k=0, 5, 1+x^(3^k))/(1-x)) \\ G. C. Greubel, Mar 29 2019
(Haskell)
a081611 n = a081611_list !! n
a081611_list = scanl1 (+) a039966_list
(Magma) R<x>:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (&*[1+x^(3^k): k in [0..5]])/(1-x) )); // G. C. Greubel, Mar 29 2019
(Sage) (product(1+x^(3^k) for k in (0..5))/(1-x)).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Mar 29 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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