A193260 G.f.: x+x^2 = Sum_{n>=1} x^n * ((1+x+x^2)^n - x^(2*n)) / (1+x+x^2)^a(n).

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, 38, 43, 44, 45, 47, 48, 49, 51, 52, 53, 56, 57, 58, 60, 61, 62, 64, 65, 66, 69, 70, 71, 73, 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

Offset: 1

Author

Paul D. Hanna, Jul 19 2011

Keywords

• nonn

Examples

			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) +...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A193259, A054861.

Programs

Mathematica

Table[n+Floor[Log[3,n]]+IntegerExponent[n!,3],{n,90}] (* Harvey P. Dale, Oct 10 2012 *)

PARI

{a(n)=if(n<1,0,n + floor(log(n+1/2)/log(3)) + valuation(n!,3))}

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)))}

Formula

a(n) = n + floor(log_3(n)) + A054861(n) for n>=1, where A054861(n) = highest power of 3 dividing n!.