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

Author

Steven Finch, May 15 2020

Keywords

• nonn

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.

Examples

			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.
		

Links

• Steven Finch, Variance of longest run duration in a random bitstring, arXiv:2005.12185 [math.CO], 2020.

Crossrefs

Cf. A119706.

Programs

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]

Formula

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