A394758 A sequence constructed by greedily sampling the Hermite distribution, with both parameters as 1, to minimize discrepancy.

2, 3, 4, 0, 1, 5, 6, 2, 3, 4, 0, 1, 2, 7, 5, 3, 2, 4, 0, 1, 8, 3, 2, 6, 4, 0, 1, 2, 5, 3, 4, 2, 0, 1, 3, 9, 2, 5, 4, 6, 0, 1, 3, 2, 7, 4, 2, 3, 0, 1, 5, 2, 4, 3, 0, 1, 2, 6, 5, 3, 4, 2, 0, 1, 10, 3, 2, 4, 7, 0, 1, 2, 5, 3, 6, 4, 2, 0, 1, 3, 8, 2, 4, 5, 0, 1, 3

Offset

1,1

Comments

The distribution is given by the PDF p(k) = (Sum_{i=0..floor(k/2)} 1/(i!*(k-2*i)!)) / e^2 = 2F0([-k/2, (1-k)/2], [], 4) / (e^2 * k!). It has p(0)=p(1)=1/e^2 which is less than p(2). From p(2) onwards it is strictly decreasing.

The arithmetic mean approaches Sum_{k>=1} k * p(k) = 3 in the limit.

Links

Jwalin Bhatt, Table of n, a(n) for n = 1..10000

Wikipedia, Hermite distribution.

Example

Let p(k) denote the probability of k and c(k) denote the number of occurrences of k among the first n-1 terms; then the expected number of occurrences of k among n random terms is given by n*p(k).
We subtract the actual occurrences c(k) from the expected occurrences and pick the one with the highest value. Let f(k) = n*p(k) - c(k).
| n |  f(0)  |  f(1)  |  f(2)  |  f(3)  |  f(4)  |  f(5)  | choice |
|---|--------|--------|--------|--------|--------|--------|--------|
| 1 |  0.135 |  0.135 |  0.203 |    -   |    -   |    -   |   2    |
| 2 |  0.270 |  0.270 | -0.593 |  0.315 |    -   |    -   |   3    |
| 3 |  0.406 |  0.406 | -0.390 | -0.526 |  0.422 |    -   |   4    |
| 4 |  0.541 |  0.541 | -0.187 | -0.368 | -0.436 |  0.365 |   0    |
| 5 | -0.323 |  0.676 |  0.015 | -0.210 | -0.295 |  0.456 |   1    |
| 6 | -0.187 | -0.187 |  0.218 | -0.052 | -0.154 |  0.548 |   5    |

Mathematica

probCountDiff[j_, k_, count_]:= k*HypergeometricPFQ[{(1-j)/2, -j/2}, {}, 4]/(E^2*j!) - Lookup[count, j, 0]
samplePDF[n_] := Module[{coeffs, unreachedVal, counts, k, probCountDiffs, mostProbable},
  coeffs = ConstantArray[0, n]; unreachedVal = 2; counts = <||>;
  Do[probCountDiffs = Table[probCountDiff[i, k, counts], {i, 0, unreachedVal}];
    mostProbable = Position[probCountDiffs, Max[probCountDiffs], {1}, 1][[1, 1]] - 1;
    If[mostProbable == unreachedVal, unreachedVal++]; coeffs[[k]] = mostProbable;
    counts[mostProbable] = Lookup[counts, mostProbable, 0] + 1, {k, 1, n}]; coeffs]
A394758 = samplePDF[120]

Prog

(Python)
from mpmath import iv
from math import factorial
def prob_count_diff(j, k, count):
    hermite = sum((iv.mpf(1) / (factorial(i)*factorial(j-2*i))) for i in range(1+j//2)) / iv.exp(2)
    return k*hermite - count
def sample_hermite_distribution(num_coeffs):
    coeffs, unreached_val, counts = [], 2, {}
    for k in range(1, num_coeffs+1):
        prob_count_diffs = [prob_count_diff(i, k, counts.get(i, 0)) for i in range(unreached_val+1)]
        most_probable = prob_count_diffs.index(max(prob_count_diffs))
        unreached_val += most_probable == unreached_val
        coeffs.append(most_probable)
        counts[most_probable] = counts.get(most_probable, 0) + 1
    return coeffs
A394758 = sample_hermite_distribution(120)

Crossrefs

Cf. A381617, A382961, A383899, A394562, A394733.

Sequence in context: A230431 A384002 A394562 * A190886 A257845 A162593

Adjacent sequences: A394755 A394756 A394757 * A394759 A394760 A394761

Keyword

nonn

Author

Jwalin Bhatt, Mar 31 2026

Status

approved