A078266 Integer part of the arithmetic mean of all the distinct numbers formed by permuting the digits of concatenation of numbers from 1 to n.

1, 16, 222, 2777, 33333, 388888, 4444444, 49999999, 555555555, 46464646464, 4102564102563, 377777777777777, 35947712418300653, 3508771929824561403, 349206349206349206348, 35265700483091787439613, 3599999999999999999999999

Offset: 1

Author

Amarnath Murthy, Nov 24 2002

Keywords

• base
• nonn

Comments

For n < 10 there are n! distinct numbers.

Examples

			a(3) = floor((123 + 132 + 213 + 231 + 312 + 321)/6) = 222;
a(4) = floor((1234 + 1243 + 1324 + 1342 + 1423 + 1432 + ... + 4312 + 4321)/24) = 66660/24 = 2777.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..369

Crossrefs

Cf. A071268, A078265.

Programs

Maple

a:= proc(n) local s, t, l;
      s:= cat("", seq(i, i=1..n)); t:= length(s);
      l:= (p-> seq(coeff(p, x, i), i=0..9))(add(x^parse(s[i]), i=1..t));
      floor((10^t-1)/9*add(i*l[i+1], i=1..9)/t)
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 05 2019

PARI

{ a(n) = c=vector(10); for(i=1,n, s=eval(Vec(Str(i))); for(j=1,#s,c[s[j]+1]++); ); l=sum(j=1,10,c[j]); (10^l-1)/9*sum(j=1,10,(j-1)*c[j])\l } \\ Max Alekseyev

Python

def A078266(n):
    s = ''.join(str(i) for i in range(1,n+1))
    return sum(int(d) for d in s)*(10**len(s)-1)//(9*len(s)) # Chai Wah Wu, Jan 04 2019

Formula

a(n) = A007953(A007908(n))*(10^A055642(A007908(n))-1)/(9*A055642(A007908(n))). - Chai Wah Wu, Jan 06 2019

Extensions

More terms from Max Alekseyev, Jan 24 2012