A138288 -id:A138288 - cp's OEIS frontend

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

Owen M Sheff, Dec 20 2018

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.

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

Cf. A255372, A138288, A322052, A004189.

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

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

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

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=', ')

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

A165293 Inverse of A038303, and generalization of A130595.

Original entry on oeis.org

1, 10, -1, 100, -20, 1, 1000, -300, 30, -1, 10000, -4000, 600, -40, 1, 100000, -50000, 10000, -1000, 50, -1, 1000000, -600000, 150000, -20000, 1500, -60, 1, 10000000, -7000000, 2100000, -350000, 35000, -2100, 70

Offset: 1

Mark Dols, Sep 13 2009

Rows sum up to A001019 (powers of 9), diagonals to A004189, a generalization of A010892 (the inverse Fibonacci). Ratio of diagonal sums converges to a decimal sequence: A000108 (Catalan numbers), which is the squared difference of sqrt(2) and sqrt(3), or 5-sqrt(24). Ratio between first binomial transform (A054320 and A138288)of A004189, converges to sqrt(2/3). 1/(2*sqrt(24)) gives A000984 (central binomial coefficients) as a decimal sequence.

Triangle T(n,k), read by rows, given by [10,0,0,0,0,0,0,0,...] DELTA [ -1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2009

			Triangle begins:
      1;
     10,    -1;
    100,   -20,   1;
   1000,  -300,  30,  -1;
  10000, -4000, 600, -40, 1;

Cf. A007318, A130595, A038303, A004189, A010892, A001079, A054320, A138288, A041041, A000108.

Sum_{k=0..n} T(n,k)*x^k = (10-x)^n. - Philippe Deléham, Dec 15 2009

G.f.: x*y/(1-10*x+x*y). - R. J. Mathar, Aug 11 2015

cp's OEIS Frontend

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

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