A334833 Total length squared of longest runs of 1's in all bitstrings of length n.

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

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*k-1)*(1/(1-2*x) - (1-x^k)/(1-2*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