

A334833


Total length squared of longest runs of 1's in all bitstrings of length n.


1



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

1,2


COMMENTS

a(n) divided by 2^n is the expected value of the longest run, squared, of heads in n tosses of a fair coin.


LINKS

Table of n, a(n) for n=1..32.
Steven Finch, Variance of longest run duration in a random bitstring, arXiv:2005.12185 [math.CO], 2020.


FORMULA

O.g.f.: Sum_{k>=1} (2*k1)*(1/(12*x)  (1x^k)/(12*x+x^(k+1))).


EXAMPLE

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.


MATHEMATICA

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]


CROSSREFS

Cf. A119706.
Sequence in context: A009147 A012593 A276235 * A048476 A122678 A256569
Adjacent sequences: A334830 A334831 A334832 * A334834 A334835 A334836


KEYWORD

nonn


AUTHOR

Steven Finch, May 15 2020


STATUS

approved



