A393161 A sequence constructed by greedily sampling the Hermite distribution, with both parameters as 1, to minimize ratio discrepancy. The distribution indexing is adjusted to be strictly decreasing and support all positive integers.

1, 2, 3, 1, 4, 2, 3, 1, 5, 2, 1, 3, 4, 1, 2, 3, 6, 1, 2, 5, 4, 1, 3, 2, 1, 3, 4, 1, 2, 5, 1, 3, 7, 2, 1, 4, 3, 2, 6, 1, 2, 3, 1, 5, 4, 1, 2, 3, 1, 2, 4, 3, 1, 5, 2, 1, 3, 6, 4, 1, 2, 3, 1, 2, 5, 1, 4, 3, 8, 7, 2, 1, 3, 2, 1, 4, 5, 1, 3, 2, 6, 1, 2, 4, 3, 1, 2

Offset

1,2

Comments

The original distribution, which supports all integers including 0, is given by 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 hence we can shift and adjust the distribution to only consider that.

The Hermite distribution used here is conditioned on being >=2, and then subtracting 1.

The distribution is given by PDF p(k) = (Sum_{i=0..ceiling(k/2)} 1/(i!*(k+1-2*i)!)) / (e^2-2) = 2F0([-(k+1)/2, -k/2], [], 4) / ((e^2 - 2)*(k+1)!).

The arithmetic mean approaches Sum_{k>=1} k * p(k) = 2.9278062629... in the limit.

The geometric mean approaches Product_{k>=2} k ^ p(k) = 2.38445049699... in the limit.

This sequence is generated by the D'Hondt (or Jefferson) apportionment method for the given distribution (choosing the smallest element in case of ties). - Pontus von Brömssen, Mar 30 2026

Links

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

Wikipedia, D'Hondt method.

Wikipedia, Hermite distribution.

Example

Let p(k) denote the probability of k and c(k) denote the count of occurrences of k so far.
We take the ratio of the actual occurrences c(k)+1 to the probability and pick the one with the lowest value. In case of ties pick the smallest k.
Since p(k) is monotonic decreasing, we only need to compute c(k) once we see c(k-1).
| n | (c(1)+1)/p(1) | (c(2)+1)/p(2) | (c(3)+1)/p(3) | (c(4)+1)/p(4) | choice |
|---|---------------|---------------|---------------|---------------|--------|
| 1 |     3.592     |       -       |       -       |       -       |   1    |
| 2 |     7.185     |     4.619     |       -       |       -       |   2    |
| 3 |     7.185     |     9.238     |     5.173     |       -       |   3    |
| 4 |     7.185     |     9.238     |    10.346     |     7.983     |   1    |
| 5 |    10.778     |     9.238     |    10.346     |     7.983     |   4    |

Mathematica

pdf[i_] := HypergeometricPFQ[{-(i+1)/2, -i/2}, {}, 4] / (i+1)!
samplePDF[numCoeffs_] := Module[
  {coeffs = {}, counts = {0}, minTime, minIndex, time},
Do[
    minTime = Infinity;
    Do[
      time = (counts[[i]] + 1)/pdf[i];
      If[time < minTime, minIndex = i; minTime = time],
      {i, 1, Length[counts]}
    ];
    If[minIndex == Length[counts], AppendTo[counts, 0]];
    counts[[minIndex]] += 1;
    AppendTo[coeffs, minIndex],
    {numCoeffs}
  ];
  coeffs
]
A393161 = samplePDF[120]

Prog

(Python)
from math import factorial
from fractions import Fraction
def hermite(k):
    return sum(Fraction(1, (factorial(i)*factorial(k+1-2*i))) for i in range(1+(k+1)//2))
def sample_hermite_distribution(num_coeffs):
  coeffs, counts = [], [0]
  for _ in range(num_coeffs):
    min_time = float('infinity')
    for i, count in enumerate(counts, start=1):
      time = (count+1) / hermite(i)
      if time < min_time:
        min_index, min_time = i, time
    if min_index == len(counts):
      counts.append(0)
    counts[min_index-1] += 1
    coeffs.append(min_index)
  return coeffs
A393161 = sample_hermite_distribution(120)

Crossrefs

Cf. A072334, A386016, A393100, A393663.

Sequence in context: A152547 A083906 A361136 * A160541 A377822 A387708

Adjacent sequences: A393158 A393159 A393160 * A393162 A393163 A393164

Keyword

nonn

Author

Jwalin Bhatt, Mar 03 2026

Status

approved