A378564 a(n) is the number of n-digit nonnegative integers with the median of the digits equal to one of the digits.

10, 9, 900, 1665, 90000, 232710, 9000000, 29055165, 900000000, 3413319138, 90000000000, 386095933170, 9000000000000, 42568084276236, 900000000000000, 4607838122919165, 90000000000000000, 491998811785538730, 9000000000000000000, 51983526276872387430, 900000000000000000000, 5447302810160797285236

Offset: 1

Author

Stefano Spezia, Dec 01 2024

Keywords

• nonn
• base

Examples

			From _David A. Corneth_, Dec 03 2024: (Start)
a(3) = 900 as every positive integer between (inclusive) 100 and 999 contains its median. The median is the middle digit after sorting which is in the digits.
a(4) = 1665. For example 2558 has digits sorted and the median, 5 is in the digits of 2558 and any permutation of digits of 2558. There are 12 such permutations so 2558 contributes 12 towards the total of a(4).
0258 has digits sorted (but a leading 0) and has 24 permutations. To account for the leading 0 we remove it and deduce the number of permutations from what is left, namely 258. That has 6 permutations. So in total 0258 adds 24 - 6 = 18 towards the total of a(4). (End)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..47 (terms 1..27 from David A. Corneth)
• David A. Corneth, PARI program

Crossrefs

Cf. A063945, A179239, A378560.

Programs

Mathematica

a[n_]:=If[OddQ[n], KroneckerDelta[n,1]+9*10^(n-1), Module[{c=0}, For[k=10^(n-1), k<=10^n-1, k++, If[MemberQ[digits=IntegerDigits[k], Median[digits]], c++]]; c]]; Array[a, 7]

PARI

\\ See Corneth link

Python

from math import prod, factorial
from itertools import combinations_with_replacement
from collections import Counter
def A378564(n):
    if n==1: return 10
    if n&1: return 9*10**(n-1)
    c, f = 0, factorial(n-1)
    for p in combinations_with_replacement(range(10),n):
        if max(p):
            a = sorted(p)
            b = a[len(a)-1>>1]+a[len(a)>>1]
            if b&1^1 and b>>1 in p:
                v = Counter(d for d in p if d).values()
                s = sum(v)
                q = prod((factorial(i) for i in v))*factorial(n-s)
                c += sum(f*i//q for i in v)
    return c # Chai Wah Wu, Dec 14 2024

Formula

a(2*n-1) = 9*10^(n-1) with a(1) = 10.

a(n) = A063945(n) for n odd.

Extensions

More terms from David A. Corneth, Dec 03 2024