A288346 -id:A288346 - cp's OEIS frontend

A288347 Median of X^2 + Y^2 where X and Y are independent random variables with B(n, 1/2) distributions.

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

Offset: 1

Matt Frank, Jun 08 2017

nonn

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.
From Robert Israel, Oct 04 2017: (Start)
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.
All terms are in A001481.
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)

Robert Israel, Table of n, a(n) for n = 1..800

Cf. A288416, which is similar, with shifted coordinates; and also A288346, which is multiplicative rather than additive.

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:
map(f, [$1..80]); # Robert Israel, Oct 04 2017

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}]

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

Original entry on oeis.org

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

Offset: 1

Matt Frank, Jun 09 2017

nonn

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.
The mean of (2X-n)^2 + (2Y-n)^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 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).

			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 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.

Cf. A288347, which is similar, with shifted coordinates; and also A288346.

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}]

cp's OEIS Frontend

A288347 Median of X^2 + Y^2 where X and Y are independent random variables with B(n, 1/2) distributions.

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