A378836 a(n) is the number of n-digit nonnegative integers with the median of the digits equal to the digital root.

10, 1, 131, 474, 10233, 50844, 1001250, 5225775, 99980565, 536333508, 9998984322, 55188464010, 999994914558, 5683515922236, 100001648752524, 585428890525092, 10000105972653645, 60302140270087340, 1000004027662440330, 6207976859006478708, 100000111315410065850

Offset: 1

Author

Stefano Spezia, Dec 09 2024

Keywords

• nonn
• base

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..43

Crossrefs

Cf. A010888, A378835.

Cf. A378560, A378564, A378837, A378838.

Programs

Mathematica

A010888[n_]:=If[n==0,0,n - 9*Floor[(n-1)/9]]; a[n_]:=If[n==1,10,Module[{c=0}, For[k=10^(n-1), k<=10^n-1, k++, If[Median[IntegerDigits[k]]==A010888[k], c++]]; c]]; Array[a, 6]

Python

from math import prod, factorial
from collections import Counter
from sympy.utilities.iterables import partitions
def A378836(n):
    if n==1: return 10
    c, f = 0, factorial(n-1)
    for i in range(1,9*n+1):
        for s,p in partitions(i,m=n,k=9,size=True):
            a = sorted(list(Counter(p).elements())+[0]*(n-s))
            b = a[len(a)-1>>1]+a[len(a)>>1]
            if b&1^1 and b>>1 == 1+(i-1)%9:
                v = list(p.values())
                p = prod((factorial(i) for i in v))*factorial(n-s)
                c += sum(f*i//p for i in v)
    return c # Chai Wah Wu, Dec 12 2024

Extensions

a(11)-a(21) from Chai Wah Wu, Dec 12 2024