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A277828
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Least number of tosses of a fair coin needed to have an even chance or better of getting a run of at least m consecutive heads or consecutive tails.
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0
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1, 2, 5, 11, 23, 45, 90, 179, 357, 712, 1422, 2842, 5681, 11360, 22716, 45430, 90856, 181709, 363413, 726822, 1453640, 2907276, 5814546, 11629086, 23258166, 46516327, 93032647
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refs;
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history;
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internal format)
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OFFSET
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1,2
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COMMENTS
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There are a family of sequences that represent the number of sequences of tosses of a fair coin to n tosses where there are no runs of m or more consecutive heads or consecutive tails. Some are given in this Encyclopedia. Their general form is given as part of the formula below. As n increases, the proportion of sequences of tosses that meet this condition decreases. When that proportion becomes a half or less of the total number of sequences of tosses, there is an even or better chance that a run of m consecutive heads or m consecutive tails occurs.
There is actually a family of sequences of which the above sequence is an instance: those in which, for successive values of m, r*g(n) <= 2^n for r > 1.
a(n) - ceiling((log 2)*2^n + (1-log 2)*n + (log 2)/2-2) equals 0 or (almost never) 1 for all n. Obtained using Weisstein's exact formula for Fibonacci k-step number seeing that the function g(N) described in the Formula section is 2*A092921(n-1,N+1). - Andrey Zabolotskiy, Nov 01 2016
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REFERENCES
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Marcus du Sautoy, The Number Mysteries, Fourth Estate, 2011, pages 126 - 127.
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LINKS
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FORMULA
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For successive integers m, where g(n) is the number of sequences of tosses of a fair coin with runs of fewer than m consecutive heads or tails out of all possible sequences of tosses to n tosses, g(n) = 2^n where n <= m-1, and thereafter g(n) = g(n-1) + g(n-2) + ... + g(n-m+1) and a(m) = the least value of n for which 2g(n) <= 2^n.
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MAPLE
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a:= proc(n) option remember; local l, j; Digits:= 50;
if n<3 then n else l:= 0$n;
for j from 0 while l[n]<1/2 do l:= seq(
(`if`(i=1, 1.0, l[i-1])+l[n-1])/2, i=1..n)
od; j
fi
end:
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MATHEMATICA
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a[n_] := a[n] = Module[{l, j}, If[n < 3, n, l = Table[0, {n}]; For[j = 0, l[[n]] < 1/2, j++, l = Table[(If[i == 1, 1, l[[i - 1]]] + l[[n - 1]])/2, {i, n}]]; j]];
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PROG
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(Python)
def a(m):
if m == 1:
return 1
g = [2**i for i in range(1, m)]
sg, lim, n = sum(g), 2**(m-1), m
while True:
g.append(sg)
sg <<= 1
sg -= g.pop(0)
if g[-1] <= lim:
return n
lim <<= 1
n += 1
print([a(i) for i in range(1, 15)])
(PARI) step(v)=my(n=#v); concat([sum(i=1, n-1, v[i])], concat(vector(n-2, i, v[i]), 2*v[n]+v[n-1]))
a(n)=if(n<3, return(n)); my(v=vector(n), flips=1, needed=1/2); v[1]=1; while(v[n]<needed, v=step(v); needed<<=1; flips++); flips \\ Charles R Greathouse IV, Nov 02 2016
(PARI) a(n)=if(n<3, return(n)); my(M=2^(n-1), v=powers(2, n-1)[2..n], i=1, m=n); while(1, v[i]=vecsum(v); if(v[i]<=M, return(m)); if(i++>#v, i=1); M*=2; m++) \\ Charles R Greathouse IV, Nov 02 2016
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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