A159062 Nearest integer to the variance of the number of tosses of a fair coin required to obtain at least n heads and n tails.

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, 25, 26, 27, 27, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 49, 50, 51, 52, 53, 53, 54, 55, 56, 57, 57, 58, 59, 60, 61, 61, 62

Offset: 1

Author

Martin Griffiths, Apr 04 2009

Keywords

• easy
• nonn

Comments

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

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

References

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

Links

• M. Griffiths, How many children?, Math. Gaz., 90 (2006), 146-149.

Crossrefs

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.

Programs

Mathematica

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]

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

Formula

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.

Extensions

More terms from Robert G. Wilson v, Apr 05 2009

Formula clarified by the author, Apr 06 2009