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Search: id:A059943
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| A059943 |
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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.). |
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+0
4
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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.
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REFERENCES
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Micael Hochster in sci.math and sci.stat.math quoting from Stochastic Processes by Sheldon Ross.
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FORMULA
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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).
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EXAMPLE
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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).
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CROSSREFS
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Cf. A007931, A059941, A059942.
Sequence in context: A064851 A134458 A009279 this_sequence A112336 A028390 A036500
Adjacent sequences: A059940 A059941 A059942 this_sequence A059944 A059945 A059946
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KEYWORD
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nice,nonn
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Feb 14 2001
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