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A334833
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Total length squared of longest runs of 1's in all bitstrings of length n.
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1
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1, 6, 21, 61, 158, 386, 902, 2051, 4565, 10006, 21668, 46484, 98958, 209360, 440627, 923299, 1927456, 4010730, 8322242, 17226050, 35578192, 73339778, 150918130, 310073773, 636173403, 1303554560, 2667935114, 5454522188, 11140674850, 22733861902, 46352349432, 94435176992
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OFFSET
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1,2
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COMMENTS
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a(n) divided by 2^n is the expected value of the longest run, squared, of heads in n tosses of a fair coin.
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LINKS
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FORMULA
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O.g.f.: Sum_{k>=1} (2*k-1)*(1/(1-2*x) - (1-x^k)/(1-2*x+x^(k+1))).
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EXAMPLE
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a(3)=21 because for the 8(2^3) possible runs 0 is longest run of heads once, 1 four times, 2 two times and 3 once and 0*1+1*4+4*2+9*1 = 21.
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MATHEMATICA
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nn = 10; Drop[Apply[Plus, Table[CoefficientList[Series[(2 n - 1) (1/(1 - 2 x) - (1 - x^n)/(1 - 2 x + x^(n + 1))), {x, 0, nn}], x], {n, 1, nn}]], 1]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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