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A288416 Median of (2X-n)^2 + (2Y-n)^2 where X and Y are independent random variables with B(n, 1/2) distributions.

%I A288416 #21 May 06 2022 13:13:51
%S A288416 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,
%T A288416 36,34,40,34,40,50,40,50,52,50,52,50,52,50,52,58,64,58,64,58,68,58,68,
%U A288416 74,68,74,72,74,72,82,80,82,80,82,80,82,80,90,100,90,100
%N A288416 Median of (2X-n)^2 + (2Y-n)^2 where X and Y are independent random variables with B(n, 1/2) distributions.
%C A288416 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 squared-distance from the origin after n pairs of coin flips.
%C A288416 The mean of (2X-n)^2 + (2Y-n)^2 is 2n, or A005843.
%C A288416 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 squared-distance for this continuous analog is n log(4). We also observe empirically that for this discrete sequence, a(n)/n approaches log(4).
%e A288416 For n=3 the probabilities of ending up at the lattice points in [-3,3]x[-3,3] are 1/64 of:
%e A288416 1 0 3 0 3 0 1
%e A288416 0 0 0 0 0 0 0
%e A288416 3 0 9 0 9 0 3
%e A288416 0 0 0 0 0 0 0
%e A288416 3 0 9 0 9 0 3
%e A288416 0 0 0 0 0 0 0
%e A288416 1 0 3 0 3 0 1
%e A288416 So the squared-distance is 2 with probability 36/64, 10 with probability 24/64, and 18 with probability 4/64; the median squared-distance is therefore 2.
%t A288416 Shifted[x_, n_] := (2 x - n)^2;
%t A288416 WeightsMatrix[n_] := Table[Binomial[n, i] Binomial[n, j], {i, 0, n}, {j, 0, n}]/2^(2 n);
%t A288416 ValuesMatrix[n_, f_] := Table[f[i, n] + f[j, n], {i, 0, n}, {j, 0, n}];
%t A288416 Distribution[n_, f_] := EmpiricalDistribution[Flatten[WeightsMatrix[n]] -> Flatten[ValuesMatrix[n, f]]];
%t A288416 NewMedian[n_, f_] :=
%t A288416 Mean[Quantile[Distribution[n, f], {1/2, 1/2 + 1/2^(2 n)}]];
%t A288416 Table[NewMedian[n, Shifted], {n, 66}]
%Y A288416 Cf. A288347, which is similar, with shifted coordinates; and also A288346.
%K A288416 nonn
%O A288416 1,1
%A A288416 _Matt Frank_, Jun 09 2017