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%I #9 Oct 10 2012 11:05:47
%S 1,2,5,6,7,9,10,11,15,16,17,19,20,21,23,24,25,28,29,30,32,33,34,36,37,
%T 38,43,44,45,47,48,49,51,52,53,56,57,58,60,61,62,64,65,66,69,70,71,73,
%U 74,75,77,78,79,83,84,85,87,88,89,91,92,93,96,97,98,100,101,102,104,105,106,109,110,111,113,114,115,117,118,119,125,126,127,129,130,131,133,134,135
%N G.f.: x+x^2 = Sum_{n>=1} x^n * ((1+x+x^2)^n - x^(2*n)) / (1+x+x^2)^a(n).
%H Harvey P. Dale, <a href="/A193260/b193260.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = n + floor(log_3(n)) + A054861(n) for n>=1, where A054861(n) = highest power of 3 dividing n!.
%e G.f.: x+x^2 = x*((1+x+x^2) - x^2)/(1+x+x^2) + x^2*((1+x+x^2)^2 - x^4)/(1+x+x^2)^2 + x^3*((1+x+x^2)^3 - x^6)/(1+x+x^2)^5 + x^4*((1+x+x^2)^4 - x^8)/(1+x+x^2)^6 + x^5*((1+x+x^2)^5 - x^10)/(1+x+x^2)^7 + x^6*((1+x+x^2)^6 - x^12)/(1+x+x^2)^9 + x^7*((1+x+x^2)^7 - x^14)/(1+x+x^2)^10 + x^8*((1+x+x^2)^8 - x^16)/(1+x+x^2)^11 + x^9*((1+x+x^2)^9 - x^18)/(1+x+x^2)^15 +...+ x^n*((1+x+x^2)^n - x^(2*n))/(1+x+x^2)^a(n) +...
%t Table[n+Floor[Log[3,n]]+IntegerExponent[n!,3],{n,90}] (* _Harvey P. Dale_, Oct 10 2012 *)
%o (PARI) {a(n)=if(n<1,0,n + floor(log(n+1/2)/log(3)) + valuation(n!,3))}
%o (PARI) {a(n)=if(n<1,0,if(n==1,1,polcoeff(sum(m=1,n+1,x^m*((1+x+x^2)^m-x^(2*m))/(1+x+x^2 +x^2*O(x^n))^if(m>=n,1,a(m)))+x^(n+1),n+1)))}
%Y Cf. A193259, A054861.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 19 2011
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