A288346 Median of 2^X + 2^Y where X and Y are independent random variables with B(n,1/2) distributions.

3, 4, 6, 9, 12, 20, 24, 40, 64, 80, 128, 160, 256, 320, 528, 768, 1088, 1536, 2176, 3072, 4352, 6144, 9216, 12288, 18432, 32768, 36864, 65536, 73728, 131072, 163840, 264192, 327680, 532480, 655360, 1064960, 1310720, 2162688, 2621440, 4325376, 6291456, 8650752

Offset: 1

Author

Matt Frank, Jun 08 2017

Keywords

• nonn

Comments

Interpretation: Start with a portfolio of stocks A and B each worth $1, and flip a pair of coins. Stock A doubles if the first coin is heads and otherwise stays constant. Stock B doubles if the second coin is heads and otherwise stays constant. This sequence gives your median portfolio value 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 2^X + 2^Y is 2(3/2)^n.

Links

• Robert Israel, Table of n, a(n) for n = 1..5299

Crossrefs

Cf. A288347, A288416, which are additive rather than multiplicative.

Programs

Maple

f:= proc(n)
local PX, i, pt, j;
for i from 0 to n do PX[i]:= binomial(n,i)/2^n od:
pt:= 0:
for j from 0 while pt^2 < 1/2 do pt:= pt + PX[j] od:
j:= j-1:
pt:= (pt-PX[j])^2:
for i from 0 do
  pt:= pt + 2*PX[j]*PX[i];
  if pt = 1/2 then error("Probability 1/2 for i=%1 j=%2",i,j) fi;
  if pt > 1/2 then return(2^i + 2^j) fi
od:
end proc:
map(f, [$1..60]); # Robert Israel, Jun 21 2017

Mathematica

TwoToThe[x_] := 2^x;
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, TwoToThe], {n, 42}]