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A288347
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Median of X^2 + Y^2 where X and Y are independent random variables with B(n, 1/2) distributions.
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3
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1, 2, 5, 9, 13, 20, 25, 34, 41, 52, 61, 74, 85, 100, 116, 130, 149, 164, 185, 202, 225, 244, 269, 290, 317, 340, 369, 394, 425, 452, 485, 520, 549, 585, 617, 653, 689, 730, 765, 808, 845, 890, 929, 976, 1017, 1066, 1109, 1160, 1205, 1258, 1305, 1360, 1409
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OFFSET
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1,2
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COMMENTS
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Interpretation: Start at the origin, and flip a pair of coins. Move right one unit if the first coin is heads, and otherwise stay in place. Then move up one unit if the second coin is heads, and otherwise stay in place. This sequence gives your median squared-distance from the origin after n pairs of coin flips.
Although a median of integers can be a half-integer, as an empirical observation only integers appear in this sequence.
The mean of X^2 + Y^2 is (n^2+n)/2, or A000217.
To avoid the possibility of half-integer values, the median can be taken as the least integer v such that Probability(X^2 + Y^2 <= v) >= 1/2.
Using the Central Limit Theorem, 4*(X^2+Y^2)/n has approximately a noncentral chi-square distribution with 2 degrees of freedom and noncentrality parameter 2*n. Thus integral_{t=0..4*a(n)/n} exp(-n-t/2) BesselI(0,sqrt(2*n*t)) dt is approximately 1.
Since a random variable not too far from normal has median approximately mu - gamma*sigma/6 where mu, sigma and gamma are the mean, standard deviation and skewness, we should expect a(n) to be approximately n^2/2 + n/4.
(End)
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LINKS
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MAPLE
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f:= proc(n) local S, P, i, j, q;
S:= sort( [seq(seq([i, j], i=0..n), j=0..n)], (a, b) -> a[1]^2 + a[2]^2 < b[1]^2 + b[2]^2);
P:= ListTools:-PartialSums(map(t -> binomial(n, t[1])*binomial(n, t[2])/2^(2*n), S));
q:= ListTools:-BinaryPlace(P, 1/2);
if P[q] = 1/2 then S[q][1]^2 + S[q][2]^2
else S[q+1][1]^2 + S[q+1][2]^2
fi
end proc:
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MATHEMATICA
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Squared[x_] := x^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] + f[j], {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, Squared], {n, 53}]
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CROSSREFS
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Cf. A288416, which is similar, with shifted coordinates; and also A288346, which is multiplicative rather than additive.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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