A242206 Number of length n binary words which contain 00 and 01 and 10 and 11 as (possibly overlapping) contiguous subsequences.

4, 18, 54, 138, 324, 724, 1568, 3326, 6954, 14390, 29552, 60344, 122684, 248586, 502366, 1013122, 2039804, 4101532, 8238520, 16534390, 33161554, 66473198, 133189224, 266771328, 534178324, 1069385154, 2140434438, 4283561466, 8571479604, 17150008420, 34311422672

Offset

5,1

Comments

The expected wait time to see all four substrings is 19/2.

Links

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Eric Weisstein's World of Mathematics, Coin Tossing

Formula

G.f.: -2*x^5*(-2+x+2*x^2)/((2*x-1)*(x^2+x-1)*(x-1)^2). - Alois P. Heinz, May 07 2014

Example

a(5) = 4 because we have: 00110, 01100, 10011, 11001.

Mathematica

sol=Solve[{A==va (z^2+z A+z C), B==vb (z^2+z A+z C), C==vc (z^2+z B+z D), D==vd (z^2+z B+z D)}, {A, B, C, D}];
S=1/(1-2 z-A-B-C-D);
vsub={va->ua-1, vb->ub-1, vc->uc-1, vd->ud-1};
Fz[z_, ua_, ub_, uc_, ud_]=Simplify[S/.sol/.vsub];
G[z_]=Simplify[Fz[z, 1, 1, 1, 0]+Fz[z, 0, 1, 1, 1]+Fz[z, 1, 0, 1, 1] +Fz[z, 1, 1, 0, 1] -Fz[z, 1, 1, 0, 0] -Fz[z, 1, 0, 1, 0]-Fz[z, 1, 0, 0, 1]-Fz[z, 0, 1, 1, 0] -Fz[z, 0, 1, 0, 1] -Fz[z, 0, 0, 1, 1]+Fz[z, 1, 0, 0, 0]+Fz[z, 0, 1, 0, 0] +Fz[z, 0, 0, 1, 0] +Fz[z, 0, 0, 0, 1] -Fz[z, 0, 0, 0, 0]];
Drop[Flatten[CoefficientList[Series[1/(1-2z)-G[z], {z, 0, 40}], z]], 5]
CoefficientList[Series[-2x^5(-2+x+2x^2)/((2x-1)(x^2+x-1)(x-1)^2), {x, 0, 50}], x] (* Harvey P. Dale, May 30 2018 *)

Crossrefs

Cf. A242167, A242257, A242323.

Sequence in context: A182031 A212250 A229788 * A181411 A238915 A212680

Adjacent sequences: A242203 A242204 A242205 * A242207 A242208 A242209

Keyword

nonn

Author

Edward Williams and Geoffrey Critzer, May 07 2014

Status

approved