OFFSET
0,2
COMMENTS
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
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
FORMULA
a(n) + A081610(n) = n+1.
From Michael Somos, Aug 31 2006: (Start)
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)).
a(n) = a(n-1) + A039966(n). (End)
MATHEMATICA
CoefficientList[Series[Product[1+x^(3^k), {k, 0, 5}]/(1-x), {x, 0, 100}], x] (* G. C. Greubel, Mar 29 2019 *)
PROG
(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) first(n)=my(s, t); vector(n, k, t=Set(digits(k, 3)); s+=t[#t]<2) \\ Charles R Greathouse IV, Sep 02 2015
(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
-- Reinhard Zumkeller, Jan 28 2012
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Mar 23 2003
STATUS
approved