A059943 Toss a fair coin and calculate the expected time until the n-th possible finite sequence of Heads and Tails first appears (ordered by length of sequence and alphabetical order so H, T, HH, HT, TH, TT, HHH, etc.).

2, 2, 6, 4, 4, 6, 14, 8, 10, 8, 8, 10, 8, 14, 30, 16, 18, 16, 18, 20, 18, 16, 16, 18, 20, 18, 16, 18, 16, 30, 62, 32, 34, 32, 38, 32, 34, 32, 34, 36, 42, 32, 34, 36, 34, 32, 32, 34, 36, 34, 32, 42, 36, 34, 32, 34, 32, 38, 32, 34, 32, 62, 126, 64, 66, 64, 70, 64, 66, 64, 70, 72

Offset: 1

Author

Henry Bottomley, Feb 14 2001

Keywords

• nice
• nonn

Comments

Note the apparent paradox: HHHHHH, HTHTHT and HHHFFF are all equally likely to appear in six tosses of the coin (1/64) and in a long sequence each is expected to appear as a subsequence roughly as many times as the others, but the expected time for HHHHHH to first appear (126) is almost twice as long as for HHHFFF (64), with HTHTHT between the two (84). This is related to the fact that in a sequence of, say, 8 successive tosses, HHHHHH could appear as a subsequence 3 times simultaneously, HTHTHT twice but HHHTTT only once.

Examples

			a(35)=38 since the expected time from xxxHHTH to completion of xxxHHTHH is 20, from xxxHHT to completion is 30, from xxxHH to completion is 32, from xxxH to completion is 36 and from xxx to completion is 38 (xxx is an earlier subsequence, perhaps empty, which cannot contribute to completion).
		

References

• M. Gardner, Chapter 5 in Time Travel and Other Mathematical Bewilderments, W. H. Freeman, 1988, pp. 63-67.
• Michael Hochster in sci.math and sci.stat.math quoting from Stochastic Processes by Sheldon Ross.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007931, A059941, A059942.

Programs

Haskell

a059943 = (* 2) . a059942  -- Reinhard Zumkeller, Apr 03 2014

Mathematica

a[n_] := (id = Drop[ IntegerDigits[n+1, 2], 1]+1; an={}; Do[PrependTo[an, If[Take[id, k] == Take[id, -k], 1, 0]], {k, 1, Length[id]}]; 2*FromDigits[an, 2]); Table[a[n], {n, 1, 72}] (* Jean-François Alcover, Nov 21 2011 *)

Formula

a(n) = 2*A059942(n). For the n-th sequence S (e.g., the 35th is HHTHH), create the set X consisting of subsequences of S which appear both at the beginning and end of S (e.g., X={H, HH, HHTHH}), then a(n) = sum_x(2^length(x)|x is in X) (e.g., a(35)=2^1+2^2+2^5=38).