A383899 -id:A383899 - cp's OEIS frontend

A383855 The n-th term of the sequence is k after every k*(k+1)/2 occurrences of 1, with multiple values following a 1 listed in order.

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

Offset: 1

Jwalin Bhatt, May 12 2025

nonn

The frequencies of the terms follow the Yule-Simon distribution with parameter value 1. The geometric mean approaches A245254 in the limit.

			After every ((2*3)/2=3) ones we see a 2,
after every ((3*4)/2=6) ones we see a 3,
after every ((4*5)/2=10) ones we see a 4 and so on.

Jwalin Bhatt, Table of n, a(n) for n = 1..10000
Wikipedia, Yule-Simon distribution

Cf. A245254, A383899.

from itertools import islice
def beta_distribution_generator():
    num_ones, num_reached = 0, 1
    while num_ones := num_ones+1:
        yield 1
        for num in range(2, num_reached+2):
            if num_ones % (num*(num+1)//2) == 0:
                yield num
                num_reached += num == num_reached+1
A383855 = list(islice(beta_distribution_generator(), 120))

A241773 A sequence constructed by greedily sampling the Gauss-Kuzmin distribution log_2[(i+1)^2/(i^2+2*i)] to minimize discrepancy.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 5, 2, 1, 6, 1, 7, 1, 2, 3, 1, 8, 1, 9, 2, 1, 4, 1, 10, 1, 3, 2, 1, 11, 1, 2, 5, 1, 12, 1, 3, 1, 2, 4, 1, 13, 1, 2, 6, 1, 14, 1, 3, 1, 2, 15, 1, 16, 1, 2, 4, 1, 3, 1, 5, 7, 1, 2, 1, 17, 1, 2, 3, 1, 18, 1, 8, 2, 1, 4, 1, 6, 1, 2, 3, 1, 5, 1, 19

Offset: 1

Jonathan Deane, Apr 28 2014

Let the sequence be A = {a(i)}, i = 1, 2, 3,... and define p(i) =
log_2[(i+1)^2/(i^2+2*i)]. Additionally, define u(j, k) = k*p(j) - N(j, k), where N(j, k) is the number of occurrences of j in {a(i)}, i = 1,..., k-1. Refer to the first argument of u as the "index" of u. Then A is defined by a(1) = 1 and, for i > 1, a(i) = m, where m is the index of the maximal element of the set {u(j, i)}, j = 1, 2, 3,... That there is a single maximal element for all i is guaranteed by the fact that p(i) - p(j) is irrational for all i not equal to j.
Interpreting sequence A as the partial coefficients of the continued fraction expansion of a real number C, say, then C = 1.44224780173510148... which is, by construction, normal (in the continued fraction sense).
The geometric mean of the sequence equals Khintchine's constant K=2.685452001 = A002210 since the frequency of the integers agrees with the Gauss-Kuzmin distribution. - Jwalin Bhatt, Feb 11 2025

			Example from _Jwalin Bhatt_, Jun 10 2025: (Start)
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(n) from the expected occurrences and pick the one with the highest value.
| n | n*p(1) - c(1) | n*p(2) - c(2) | n*p(3) - c(3) | choice |
|---|---------------|---------------|---------------|--------|
| 1 |     0.415     |     0.169     |     0.093     |   1    |
| 2 |    -0.169     |     0.339     |     0.186     |   2    |
| 3 |     0.245     |    -0.490     |     0.279     |   3    |
| 4 |     0.660     |    -0.320     |    -0.627     |   1    |
(End)

Jonathan Deane, Table of n, a(n) for n = 1..10000
Jonathan Deane, An integer sequence whose members obey a given p.d.f.

Cf. A002210, A084580, A089618, A381617, A382961, A383238, A383899.

pdf := i -> -log[2](1 - 1/(i+1)^2);
gen_seq := proc(n)
local i, j, N, A, u, mm, ndig;
ndig := 40; N := 'N';
for i from 1 to n do    N[i] := 0;      end do;
A := 'A'; A[1] := 1; N[1] := 1;
for i from 2 to n do
        u := 'u';
        for j from 1 to n do
                u[j] := i*pdf(j) - N[j];
        end do;
        mm := max_maxind(evalf(convert(u, list), ndig));
        if mm[3] then
                A[i] := mm[1];
                N[mm[1]] := N[mm[1]] + 1;
        else
                return();
        end if;
end do;
return(convert(A, list));
end:
max_maxind := proc(inl)
local uniq, mxind, mx, i;
uniq := `true`;
if nops(inl) = 1 then return([1, inl[1], uniq]); end if;
mxind := 1; mx := inl[1];
for i from 2 to nops(inl) do
        if inl[i] > mx then
                mxind := i;
                mx := inl[i];
                uniq := `true`;
        elif inl[i] = mx then
                uniq := `false`;
        end if;
end do;
return([mxind, mx, uniq]);
end:
gen_seq(100);

probCountDiff[j_, k_, count_] := k*-Log[2, 1 - (1/((j + 1)^2))] - Lookup[count, j, 0]
samplePDF[n_] := Module[{coeffs, unreachedVal, counts, k, probCountDiffs, mostProbable},
  coeffs = ConstantArray[0, n]; unreachedVal = 1; counts = <||>;
  Do[probCountDiffs = Table[probCountDiff[i, k, counts], {i, 1, unreachedVal}];
    mostProbable = First@FirstPosition[probCountDiffs, Max[probCountDiffs]];
    If[mostProbable == unreachedVal, unreachedVal++]; coeffs[[k]] = mostProbable;
    counts[mostProbable] = Lookup[counts, mostProbable, 0] + 1; , {k, 1, n}]; coeffs]
A241773 = samplePDF[120]  (* Jwalin Bhatt, Jun 04 2025 *)

from fractions import Fraction
def prob_count_diff(j, k, count):
    return  - ((1 - Fraction(1,(j+1)*(j+1)))**k) * (2**count)
def sample_pdf(n):
    coeffs, unreached_val, counts = [], 1, {}
    for k in range(1, n+1):
        prob_count_diffs = [prob_count_diff(i, k, counts.get(i, 0)) for i in range(1, unreached_val+1)]
        most_probable = prob_count_diffs.index(max(prob_count_diffs)) + 1
        unreached_val += most_probable == unreached_val
        coeffs.append(most_probable)
        counts[most_probable] = counts.get(most_probable, 0) + 1
    return coeffs
A241773 = sample_pdf(120)  # Jwalin Bhatt, Feb 09 2025

cp's OEIS Frontend

A383855 The n-th term of the sequence is k after every k*(k+1)/2 occurrences of 1, with multiple values following a 1 listed in order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python