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

1,1

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

Table of n, a(n) for n=1..72.

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

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.

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]

Prog

(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

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.

Sequence in context: A106618 A320113 A080684 * A301274 A093616 A331172

Adjacent sequences: A159059 A159060 A159061 * A159063 A159064 A159065

Keyword

easy,nonn

Author

Martin Griffiths, Apr 04 2009

Extensions

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

Formula clarified by the author, Apr 06 2009

Status

approved