A322052 -id:A322052 - 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

A322054 Number of decimal strings of length n that do not contain a specific string xx (where x is a single digit).

Original entry on oeis.org

10, 99, 981, 9720, 96309, 954261, 9455130, 93684519, 928256841, 9197472240, 91131561729, 902961305721, 8946835807050, 88648174014939, 878355088397901, 8703029361715560, 86232460051021149, 854419404714630381, 8465866782890863770

Offset: 1

N. J. A. Sloane, Dec 21 2018

See A322053 for the number that do contain the specified string.

			Suppose the string is 00. At length 2 there are 99 strings that do not contain it. At length 3 there are 19 strings that do not contain it, 000, 00x, and x00, where x is any nonzero digit. So a(3) = 1000-19 = 981.

Robert P. P. McKone, Table of n, a(n) for n = 1..1000
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2024.
Index entries for linear recurrences with constant coefficients, signature (9,9).

Cf. A322052, A004189, A322053.

Suggested by A322628.

T[n_, k_] := LinearRecurrence[{n - 1, n - 1}, {n, n^2 - 1}, k];
T[10, {1, 19}] (* Robert P. P. McKone, Dec 31 2020 *)

G.f.: x*(10+9*x)/(1-9*x-9*x^2).

a(n) = 9*a(n-1) + 9*a(n-2) for n >= 3.

A322053 Number of decimal strings of length n that contain a specific string xx (where x is a single digit).

Original entry on oeis.org

0, 1, 19, 280, 3691, 45739, 544870, 6315481, 71743159, 802527760, 8868438271, 97038694279, 1053164192950, 11351825985061, 121644911602099, 1296970638284440, 13767539948978851, 145580595285369619, 1534133217109136230, 16117424311550552641

Offset: 1

N. J. A. Sloane, Dec 21 2018

See A322054 for the number that do not contain the specified string.

			Suppose the desired string is 00. At length 2 that is the only possibility. At length 3 there are 19 strings that contain it: 000, 00d, and d00, where d is any nonzero digit.

Cf. A322052, A004189, A322054.

Suggested by A322628.

G.f. = x^2/((1-10*x)*(1-9*x-9*x^2)).

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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A322054 Number of decimal strings of length n that do not contain a specific string xx (where x is a single digit).

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A322053 Number of decimal strings of length n that contain a specific string xx (where x is a single digit).

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Author

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Examples

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Formula