

A288416


Median of (2Xn)^2 + (2Yn)^2 where X and Y are independent random variables with B(n, 1/2) distributions.


2



2, 4, 2, 4, 10, 8, 10, 8, 10, 16, 18, 20, 18, 20, 26, 20, 26, 20, 26, 32, 26, 32, 34, 36, 34, 36, 34, 40, 34, 40, 50, 40, 50, 52, 50, 52, 50, 52, 50, 52, 58, 64, 58, 64, 58, 68, 58, 68, 74, 68, 74, 72, 74, 72, 82, 80, 82, 80, 82, 80, 82, 80, 90, 100, 90, 100
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OFFSET

1,1


COMMENTS

Interpretation: Start at the origin, and flip a pair of coins. Move right one unit if the first coin is heads, and otherwise left one unit. Then move up one unit if the second coin is heads, and otherwise down one unit. This sequence gives your median squareddistance from the origin after n pairs of coin flips.
The mean of (2Xn)^2 + (2Yn)^2 is 2n, or A005843.
A continuous analog draws each move from N(0,1) rather than from {+1,1}, so the final x and y coordinates are distributed as N(0,sqrt(n)). Then the final point has probability 1  exp(r^2/2n) of being within r of the origin, and the median squareddistance for this continuous analog is n log(4). We also observe empirically that for this discrete sequence, a(n)/n approaches log(4).


LINKS

Table of n, a(n) for n=1..66.


EXAMPLE

For n=3 the probabilities of ending up at the lattice points in [3,3]x[3,3] are 1/64 of:
1 0 3 0 3 0 1
0 0 0 0 0 0 0
3 0 9 0 9 0 3
0 0 0 0 0 0 0
3 0 9 0 9 0 3
0 0 0 0 0 0 0
1 0 3 0 3 0 1
So the squareddistance is 2 with probability 36/64, 10 with probability 24/64, and 18 with probability 4/64; the median squareddistance is therefore 2.


MATHEMATICA

Shifted[x_, n_] := (2 x  n)^2;
WeightsMatrix[n_] := Table[Binomial[n, i] Binomial[n, j], {i, 0, n}, {j, 0, n}]/2^(2 n);
ValuesMatrix[n_, f_] := Table[f[i, n] + f[j, n], {i, 0, n}, {j, 0, n}];
Distribution[n_, f_] := EmpiricalDistribution[Flatten[WeightsMatrix[n]] > Flatten[ValuesMatrix[n, f]]];
NewMedian[n_, f_] :=
Mean[Quantile[Distribution[n, f], {1/2, 1/2 + 1/2^(2 n)}]];
Table[NewMedian[n, Shifted], {n, 66}]


CROSSREFS

Cf. A288347, which is similar with shifted coordinates; and also A288346.
Sequence in context: A191370 A298242 A282283 * A240893 A241108 A151706
Adjacent sequences: A288413 A288414 A288415 * A288417 A288418 A288419


KEYWORD

nonn


AUTHOR

Matt Frank, Jun 09 2017


STATUS

approved



