%I #10 Jan 23 2019 10:53:16

%S 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,16,17,18,19,20,21,22,22,23,24,

%T 25,26,27,27,28,29,30,31,32,32,33,34,35,36,37,37,38,39,40,41,41,42,43,

%U 44,45,45,46,47,48,49,49,50,51,52,53,53,54,55,56,57,57,58,59,60,61,61,62

%N Nearest integer to the variance of the number of tosses of a fair coin required to obtain at least n heads and n tails.

%C For any n, either a(n+1)-a(n)=0 or a(n+1)-a(n)=1.

%C a(n)/b(n) tends to 1 - 2/Pi as n tends to infinity, where b(n) is the n-th term of A159061.

%D M. Griffiths, The Backbone of Pascal's Triangle, United Kingdom Mathematics Trust, 2008, pp. 68-72.

%H M. Griffiths, <a href="https://www.jstor.org/stable/3621443">How many children?</a>, Math. Gaz., 90 (2006), 146-149.

%F a(n) is the nearest integer to 2*n*(1+binomial(2*n,n)/(2^(2*n)))-((n*binomial(2*n,n))/(2^(2*n-1)))^2.

%t f[n_] := Round[2^(1 - 4 n) n (16^n + Binomial[2 n, n] (4^n - 2 n Binomial[2 n, n]))]; Array[f, 72]

%o (PARI) a(n) = round(2*n*(1+binomial(2*n,n)/(2^(2*n)))-((n*binomial(2*n,n))/(2^(2*n-1)))^2) \\ _Felix Fröhlich_, Jan 23 2019

%Y The nearest integer to the expected number of tosses of a fair coin required to obtain at least n heads and n tails is given in A159061.

%K easy,nonn

%O 1,1

%A _Martin Griffiths_, Apr 04 2009

%E More terms from _Robert G. Wilson v_, Apr 05 2009

%E Formula clarified by the author, Apr 06 2009