A106744 Given n shoelaces, each with two aglets; sequence gives number of aglet pairs that must be picked up to guarantee that the probability that no shoelace is left behind is > 1/2.

1, 1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45

Offset

1,3

Comments

Assistance in extending sequence given by Gerald McGarvey

Fieggen's site has pictures of aglets (and hints on repairing them). - N. J. A. Sloane May 16 2005

Links

Table of n, a(n) for n=1..50.

Ian Fieggen, Aglet repair

Example

a(2)=2 because given 2 shoelaces, p=1/3 that the first two aglets picked up will be on a single shoelace, requiring another pick-up and p=2/3 that they won't, so the mean no. of pick-ups is (1/3)*2 + (2/3)*1 = 4/3, for which the ceiling is 2.

Prog

(PARI) choose(n, k)=gamma(n+1)/(gamma(n-k+1)*gamma(k+1)) a(n)=ceil(solve(x=n/2, n, 2^(2*(n-x))*choose(n, 2*(n-x))-(1/2)*choose(2*n, 2*x))) for(n=3, 50, print1(a(n), ", ")) (McGarvey)

Crossrefs

See A106829 for another version.

Sequence in context: A303788 A366871 A319288 * A332613 A072932 A029915

Adjacent sequences: A106741 A106742 A106743 * A106745 A106746 A106747

Keyword

nonn

Author

Neil Fernandez, May 16 2005

Extensions

More terms from Gerald McGarvey, May 17 2005

Status

approved