A193259 G.f.: x = Sum_{n>=1} x^n * ((1+x)^n - x^n) / (1+x)^a(n).

1, 4, 5, 9, 10, 12, 13, 18, 19, 21, 22, 25, 26, 28, 29, 35, 36, 38, 39, 42, 43, 45, 46, 50, 51, 53, 54, 57, 58, 60, 61, 68, 69, 71, 72, 75, 76, 78, 79, 83, 84, 86, 87, 90, 91, 93, 94, 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

Offset: 1

Author

Paul D. Hanna, Jul 19 2011

Keywords

• nonn

Comments

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

Examples

			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) +...
		

Crossrefs

Cf. A193263, A193260, A135560, A011371.

Programs

PARI

{a(n)=if(n<1,0,n + floor(log(n+1/2)/log(2)) + valuation(n!,2))}

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)))}

Formula

a(n) = n + floor(log_2(n)) + A011371(n) for n>=1, where A011371(n) = highest power of 2 dividing n!.

a(n) = a(n-1) + A135560(n) for n>1 with a(1)=1.