A135342 -id:A135342 - cp's OEIS frontend

A382383 Number of distinct variances of nonempty subsets of {1, ..., n}.

0, 1, 2, 4, 7, 13, 23, 40, 68, 124, 208, 368, 559, 918, 1352, 2017, 2891, 4122, 5506, 7458, 9623, 12620, 16125, 20626, 25401, 31513, 38587, 47244, 56592, 68021, 80503, 95859, 112137, 131986, 153353, 178434, 205627, 236266, 269884, 307167, 346844, 394924, 445797, 501739

Offset: 0

Pontus von Brömssen, Mar 23 2025

nonn

The variance of a nonempty set X is (Sum_{x in X} (x-m)^2)/|X|, where m is the average of X and |X| is the size of X.

			For n = 4, the following a(4) = 7 variances occur for subsets of {1, 2, 3, 4}:
   variance | corresponding subsets
   ---------+----------------------
       0    | {1}, {2}, {3}, {4}
      1/4   | {1,2}, {2,3}, {3,4}
      2/3   | {1,2,3}, {2,3,4}
       1    | {1,3}, {2,4}
      5/4   | {1,2,3,4}
     14/9   | {1,2,4}, {1,3,4}
      9/4   | {1,4}

Cf. A005418, A135342, A208531, A382381, A382382.

from fractions import Fraction
def A382383_lst(n):
    s,lst=set(),[0]
    for k in range(n):
        s|={ (x+k,x2+k**2,l+1 ) for (x,x2,l) in s }
        s.add( (k,k**2,1) )
        lst.append(len({ Fraction(x2,l) - Fraction(x,l)**2 for (x,x2,l) in s }))
    return lst # Bert Dobbelaere, Apr 06 2025

a(n) <= 1 + Sum_{k=1..n-1} A005418(k). The smallest positive n for which strict inequality holds is n = 7. This is because there exist subsets of {1, ..., 7} which are not translates or reflections of each other, but nonetheless have the same variance. For example, {1,5}, {1,3,4,5,7}, and {1,2,3,4,5,6,7} all have variance 4, and {1,2,6} and {1,2,3,5,6,7} both have variance 14/3.

a(34)-a(43) from Bert Dobbelaere, Apr 06 2025

A360192 Number of distinct means of nonempty subsets of points {(x,y)|1<=x<=n, 1<=y<=n}.

Original entry on oeis.org

1, 13, 157, 1409, 8237, 35165, 120809, 351865, 903921, 2102065, 4511365, 9056021, 17194093, 31125005, 54080341, 90666045, 147314445

Offset: 1

Nitai Sasson, Jan 29 2023

			For n = 2 the a(2) = 13 means are (1, 1), (1, 2), (1, 3/2), (2, 1), (2, 2), (2, 3/2), (3/2, 1), (3/2, 2), (3/2, 3/2), (4/3, 4/3), (4/3, 5/3), (5/3, 4/3), (5/3, 5/3).
They can be visualized as follows:
Mark the set of points with "x" in a 2 X 2 matrix:
(xx)
(xx)
The points not in a particular subset are replaced with "."
For example, the subset {(1,1),(1,2)}
                   (x.)
is represented by: (x.)
The means for n=2 arise from the following subsets:
       (x.)
(1,1): (..)
-
       (.x)
(2,1): (..)
-
       (..)
(1,2): (x.)
-
       (..)
(2,2): (.x)
-
           (x.)    (.x)    (xx)
(3/2,3/2): (.x) or (x.) or (xx)
-
         (xx)
(3/2,1): (..)
-
         (..)
(3/2,2): (xx)
-
         (x.)
(1,3/2): (x.)
-
         (.x)
(2,3/2): (.x)
-
           (xx)
(4/3,4/3): (x.)
-
           (x.)
(4/3,5/3): (xx)
-
           (xx)
(5/3,4/3): (.x)
-
           (.x)
(5/3,5/3): (xx)

Two-dimensional analog of A135342.

a(10)-a(17) from Zachary DeStefano, Jan 29 2023

cp's OEIS Frontend

A382383 Number of distinct variances of nonempty subsets of {1, ..., n}.

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