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A386904 A sequence constructed by greedily sampling the Borel distribution for parameter value 1/2 to minimize discrepancy.

%I A386904 #16 Aug 22 2025 17:19:56
%S A386904 1,2,1,1,3,1,4,1,2,1,1,5,1,2,1,1,3,1,1,2,1,6,1,1,2,1,1,3,1,2,1,4,1,1,
%T A386904 2,1,1,7,1,2,1,3,1,1,8,1,2,1,1,5,1,2,1,1,3,1,1,2,1,4,1,1,2,1,3,1,1,2,
%U A386904 1,1,9,1,2,1,1,4,1,3,1,2,1,1,6,1,2,1,1,1,3,1,2,1,5,1,1,2,1,1,4,1,2,1,3,1,1,2,1,1,10
%N A386904 A sequence constructed by greedily sampling the Borel distribution for parameter value 1/2 to minimize discrepancy.
%C A386904 The geometric mean approaches A386009 in the limit.
%C A386904 The Borel distribution with parameter value 1/2 has PDF p(i) = (i/2)^(i-1) / (exp(i/2)*i!).
%H A386904 Jwalin Bhatt, <a href="/A386904/b386904.txt">Table of n, a(n) for n = 1..10000</a>
%H A386904 Wikipedia, <a href="https://en.wikipedia.org/wiki/Borel_distribution">Borel distribution</a>
%e A386904 Let p(k) denote the probability of k and c(k) denote the count of occurrences of k so far, then the expected occurrences of k at n-th step are given by n*p(k).
%e A386904 We subtract the actual occurrences c(k) from the expected occurrences and pick the one with the highest value.
%e A386904 | n | n*p(1) - c(1) | n*p(2) - c(2) | n*p(3) - c(3) | choice |
%e A386904 |---|---------------|---------------|---------------|--------|
%e A386904 | 1 |     0.606     |       -       |       -       |   1    |
%e A386904 | 2 |     0.213     |     0.367     |       -       |   2    |
%e A386904 | 3 |     0.819     |    -0.448     |     0.251     |   1    |
%e A386904 | 4 |     0.426     |    -0.264     |     0.334     |   1    |
%e A386904 | 5 |     0.032     |    -0.080     |     0.418     |   3    |
%t A386904 samplePDF[n_]:=Module[{coeffs, unreachedVal, counts, k, probCountDiffs, mostProbable},
%t A386904   coeffs=ConstantArray[0, n]; unreachedVal=1; counts=<||>;
%t A386904   Do[probCountDiffs=Table[probCountDiff[i, k, counts], {i, 1, unreachedVal}];
%t A386904     mostProbable=First@FirstPosition[probCountDiffs, Max[probCountDiffs]];
%t A386904     If[mostProbable==unreachedVal, unreachedVal++]; coeffs[[k]]=mostProbable;
%t A386904     counts[mostProbable]=Lookup[counts, mostProbable, 0]+1; , {k, 1, n}]; coeffs]
%t A386904 A386904=samplePDF[120]
%Y A386904 Cf. A386009, A386016.
%K A386904 nonn,new
%O A386904 1,2
%A A386904 _Jwalin Bhatt_, Aug 07 2025