A322628 Number of n-digit decimal numbers containing a fixed 2-digit integer with distinct digits as a substring.

0, 0, 1, 19, 279, 3671, 45431, 540639, 6260959, 71068951, 794428551, 8773216559, 95937737039, 1040604153831, 11210103801271, 120060433858879, 1279394234787519, 13573881914016311, 143459424905375591, 1511020367139739599, 15866744246492020399

Offset: 0

Author

Owen M Sheff, Dec 20 2018

Keywords

• nonn
• base
• easy

Comments

First differences of A322052. - Jon E. Schoenfield, Jul 31 2021

See A138288 for the number of n-digit decimal numbers that do not contain a fixed 2-digit integer with distinct digits as a substring.

Links

• Colin Barker, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (20,-101,10).

Crossrefs

Cf. A255372, A138288, A322052, A004189.

Programs

GAP

a:=[0,1,19];; for n in [4..20] do a[n]:=20*a[n-1]-101*a[n-2]+10*a[n-3]; od; Concatenation([0],a); # Muniru A Asiru, Dec 21 2018

Maple

seq(coeff(series(x^2*(x-1)/((10*x-1)*(x^2-10*x+1)),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Dec 21 2018

PARI

concat([0,0], Vec(x^2*(x-1)/((10*x-1)*(x^2-10*x+1)) + O(x^30))) \\ Colin Barker, Dec 21 2018

Python

def find_int(i):
  if i == 0: return (0)
  intlist = [0,1,19]
  for n in range(4,i+2):
    if n > 3:
      a = 10*(intlist[n-2])+(9*10**(n-3)-intlist[n-3])
      intlist.append(a)
  return (intlist[i-1])
for i in range(100):
  print(find_int(i), end=', ')

Formula

a(n) = 10*a(n-1) - a(n-2) + 9*10^(n-3) with a(0) = a(1) = 0, a(2) = 1.

G.f.: x^2*(x-1)/((10*x-1)*(x^2-10*x+1)). - Alois P. Heinz, Dec 20 2018

a(n) = (27*10^n + 5*(5-2*sqrt(6))^n*(-3+sqrt(6)) - 5*(3+sqrt(6))*(5+2*sqrt(6))^n) / 30 for n>0. - Colin Barker, Dec 21 2018