A178851 The number of length n sequences on {0,1,2}(ternary sequences) that contain a prime number of 2's.

0, 0, 1, 7, 32, 121, 412, 1317, 4048, 12144, 35904, 105249, 306968, 892217, 2585468, 7468532, 21500800, 61688513, 176477988, 503906221, 1438235592, 4110846808, 11789919200, 33991337521, 98657320240, 288505634480, 850146795840, 2522918119392, 7531922736384

Offset: 0

Author

Geoffrey Critzer, Dec 27 2010

Keywords

• nonn

Comments

a(n) is the number of positive integers less than 3^n that when expressed as a ternary numeral contain a prime number of 2's.

a(n)/3^n is the probability that a series of Bernoulli trials with probability of success equal to 1/3 will result in a prime number of successes. Cf. A121497

Examples

			a(3)=7 because 8,17,20,23,24,25,26 have a prime number of 2's in their ternary notation.
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 0..2000

Programs

Mathematica

P=Table[Prime[m],{m,1,200}];Range[0,20]! CoefficientList[Series[Exp[2x] Sum[x^p/p!,{p,P}],{x,0,20}],x]

Formula

E.g.f.:exp(2x)*(x^2/2!+x^3/3!+x^5/5!+...)

a(n) = Sum Binomial(n,p)*2^(n-p) where the sum is taken over the prime numbers.