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%I #13 Mar 30 2012 18:37:28
%S 1,4,5,9,10,12,13,18,19,21,22,25,26,28,29,35,36,38,39,42,43,45,46,50,
%T 51,53,54,57,58,60,61,68,69,71,72,75,76,78,79,83,84,86,87,90,91,93,94,
%U 99,100,102,103,106,107,109,110,114,115,117,118,121,122,124,125,133,134,136,137,140,141,143,144,148,149,151,152,155,156,158,159,164,165,167
%N G.f.: x = Sum_{n>=1} x^n * ((1+x)^n - x^n) / (1+x)^a(n).
%C Sum_{n>=1} 1/phi^a(n) = 1/phi + Sum_{n>=1} 1/phi^(a(n) + 2*n) = 0.880771363850914609641... where phi = (sqrt(5)+1)/2. - Paul D. Hanna, Sep 14 2011
%F a(n) = n + floor(log_2(n)) + A011371(n) for n>=1, where A011371(n) = highest power of 2 dividing n!.
%F a(n) = a(n-1) + A135560(n) for n>1 with a(1)=1.
%e G.f.: x = x*((1+x) - x)/(1+x) + x^2*((1+x)^2 - x^2)/(1+x)^4 + x^3*((1+x)^3 - x^3)/(1+x)^5 + x^4*((1+x)^4 - x^4)/(1+x)^9 + x^5*((1+x)^5 - x^5)/(1+x)^10 + x^6*((1+x)^6 - x^6)/(1+x)^12 + x^7*((1+x)^7 - x^7)/(1+x)^13 + x^8*((1+x)^8 - x^8)/(1+x)^18 +...+ x^n*((1+x)^n - x^n)/(1+x)^a(n) +...
%o (PARI) {a(n)=if(n<1,0,n + floor(log(n+1/2)/log(2)) + valuation(n!,2))}
%o (PARI) {a(n)=if(n<1,0,if(n==1,1,polcoeff(sum(m=1,n+1,x^m*((1+x)^m-x^m)/(1+x +x^2*O(x^n))^if(m>=n,1,a(m)))+x^(n+1),n+1)))}
%Y Cf. A193263, A193260, A135560, A011371.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 19 2011
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