A255372 -id:A255372 - cp's OEIS frontend

A255371 Number of strings of n decimal digits that contain at least one "0" digit that is not part of a string of two or more consecutive "0" digits.

0, 1, 18, 252, 3177, 37764, 432315, 4821867, 52767711, 569171142, 6070198824, 64154357361, 673034324472, 7017585817887, 72795938474871, 751858421307975, 7736579039166894, 79354228046171004, 811679794900979769, 8282239107946760700, 84331460977774328115

Offset: 0

Jon E. Schoenfield, Feb 21 2015

Let A(n,k) be the number of strings of n decimal digits that contain at least one string of exactly k consecutive "0" digits (i.e., at least one string of k consecutive "0" digits that is not part of a string of more than k consecutive "0" digits). This sequence gives the values of A(n,k) for k=1.

			a(1) = 1 because there is only 1 one-digit string that contains a "0" digit, i.e., "0" itself.
a(2) = 18 because there are 18 two-digit strings that contain a "0" digit that is not part of a string of two or more consecutive "0" digits; using "+" to represent a nonzero digit, the 18 strings comprise 9 of the form "0+" and 9 of the form "+0". ("00" is excluded.)
a(3) = 252 because there are 252 three-digit strings that contain at least one "0" digit that is not part of a string of two or more consecutive "0" digits; using "+" as above, the 252 strings comprise 81 of the form "0++", 81 of the form "+0+", 81 of the form "++0", and 9 of the form "0+0".

Colin Barker, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (20,-109,99,-90).

Cf. A255372-A255380 (for cases k=2 through k=10; see Comments).

LinearRecurrence[{20, -109, 99, -90}, {0, 1, 18, 252}, 30] (* Paolo Xausa, Aug 27 2024 *)

concat(0, Vec(x*(x-1)^2/((10*x-1)*(9*x^3-9*x^2+10*x-1)) + O(x^100))) \\ Colin Barker, Feb 27 2015

a(0)=0, a(1)=1, a(n) = 9*(10^(n-2) - a(n-2) + sum_{i=1..n-1} a(i)) for n>=2.
G.f.: x*(x-1)^2/((10*x-1)*(9*x^3-9*x^2+10*x-1)). - Alois P. Heinz, Feb 26 2015
a(n) = 20*a(n-1) - 109*a(n-2) + 99*a(n-3) - 90*a(n-4) for n>3. - Colin Barker, Feb 27 2015
a(n) ~ 10^n. - Stefano Spezia, Aug 28 2024

a(0)=0 prepended by Jon E. Schoenfield, Feb 21 2015

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

Original entry on oeis.org

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

A255373 Number of strings of n decimal digits that contain at least one string of exactly 3 consecutive "0" digits.

Original entry on oeis.org

0, 0, 0, 1, 18, 261, 3420, 42291, 503757, 5845383, 66525399, 745904151, 8264888316, 90700808526, 987461965116, 10678505242392, 114817381566435, 1228431892382460, 13086248073415290, 138875261344657416, 1468815363559657773, 15488131104999233505

Offset: 0

Jon E. Schoenfield, Feb 22 2015

			a(3) = 1 because there is only 1 three-digit string that contains the substring "000", i.e., "000" itself.
a(4) = 18 because there are 18 four-digit strings that contain a "000" substring that is not part of a string of four or more consecutive "0" digits; using "+" to represent a nonzero digit, the 18 strings comprise 9 of the form "000+" and 9 of the form "+000". ("0000" is excluded.)
a(5) = 261 because there are 261 five-digit strings that contain a "000" substring that is not part of a string of four or more consecutive "0" digits; using "+" as above and "." to denote any digit (0 or otherwise), the 261 strings comprise 9*10=90 of the form "000+.", 9*9=81 of the form "+000+", and 10*9=90 of the form ".+000".
a(6) = 3420 because there are 3420 six-digit strings that contain a "000" substring that is not part of a string of four or more consecutive "0" digits; using "+" and "." as above, the 3420 strings comprise 9*10*10=900 of the form "000+..", 9*9*10=810 of the form "+000+.", 10*9*9=810 of the form ".+000+", and 10*10*9=900 of the form "..+000".
a(7) = 42291 because there are 42291 seven-digit strings that contain at least one "000" substring that is not part of a string of four or more consecutive "0" digits; using "+" and "." as above, the 42291 strings comprise 9*10*10*10=9000 of the form "000+...", 9*9*10*10=8100 of the form "+000+..", 10*9*9*10=8100 of the form ".+000+.", 10*10*9*9=8100 of the form "..+000+", and 999*9=8991 that are of the form "...+000" but not of the form "000+000" (since 9 strings of that latter form were already counted among the 9000 of the form "000+...").

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (20,-100,0,-9,99,-90).

Cf. A255371, A255372 (for the k=1 and k=2 cases of "exactly k consecutive '0' digits"), A255374-A255380 (for the cases k=4 through k=10).

a(0) = a(1) = a(2) = 0, a(3) = 1, a(n) = 9*(10^(n-4) - a(n-4) + sum_{i=3..n-1} a(i)) for n>=4.

G.f.: x^3*(x-1)^2/((10*x-1)*(9*x^5-9*x^4+10*x-1)). - Alois P. Heinz, Feb 26 2015

A255381 Number of strings of k+n decimal digits that contain one string of exactly k consecutive "0" digits, where k >= n.

Original entry on oeis.org

1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, 747000000, 8280000000, 90900000000, 990000000000, 10710000000000, 115200000000000, 1233000000000000, 13140000000000000, 139500000000000000, 1476000000000000000, 15570000000000000000, 163800000000000000000

Offset: 0

Jon E. Schoenfield, Feb 27 2015

This sequence gives the first k+1 nonzero terms from A255371 (the k=1 case), A255372 (the k=2 case), etc., through A255380 (the k=10 case). Those sequences' definitions concern the number of strings of decimal digits that contain "at least one" string of exactly k consecutive "0" digits; the present sequence omits the words "at least" because, with k >= n, and thus 2k >= k+n, it is not possible to have more than one string of exactly k consecutive "0" digits in a string of k+n digits. (Two strings each having exactly k consecutive "0" digits would have to be separated by at least one nonzero digit, or else they would constitute a single string of exactly 2k consecutive "0" digits.)
Omitting the zero terms of each, A255371 through A255380 begin
1, 18, 252, 3177, 37764, 432315, 4821867, 52767711, ...
1, 18, 261, 3411, 42057, 499383, 5775480, 65506986, ...
1, 18, 261, 3420, 42291, 503757, 5845383, 66525399, ...
1, 18, 261, 3420, 42300, 503991, 5849757, 66595383, ...
1, 18, 261, 3420, 42300, 504000, 5849991, 66599757, ...
1, 18, 261, 3420, 42300, 504000, 5850000, 66599991, ...
1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ...
1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ...
1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ...
1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ...
and the terms of this present sequence give the limiting value for each column.

			Trivially, a(0)=1 because there is 1 string of k decimal digits that contains one string of exactly k consecutive "0" digits, where k >= 0: namely, the string of k consecutive "0" digits itself.
a(1)=18 because there are 18 strings of k+1 decimal digits that contain one string of exactly k consecutive "0" digits, where k >= 1. Letting "S" and "+" represent the string of exactly k consecutive "0" digits and any nonzero digit, respectively, the 18 strings comprise 9 of the form "S+" and 9 of the form "+S".
a(2)=261 because there are 261 strings of k+2 decimal digits that contain one string of exactly k consecutive "0" digits, where k >= 2. Letting "S", "+", and "." represent the string of exactly k consecutive "0" digits, any nonzero digit, and any digit (zero or nonzero), respectively, the 261 strings comprise 9*10=90 of the form "S+.", 9*9=81 of the form "+S+", and 10*9=90 of the form ".+S".
a(3)=3420 because there are 3420 strings of k+3 decimal digits that contain one string of exactly k consecutive "0" digits, where k >= 3. Using "S", "+", and "." as above, the 3420 strings comprise 9*10*10=900 of the form "S+..", 9*9*10=810 of the form "+S+.", 10*9*9=810 of the form ".+S+", and 10*10*9=900 of the form "..+S".

Colin Barker, Table of n, a(n) for n = 0..996
Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A255371, A255372, A255373, A255374, A255375, A255376, A255377, A255378, A255379, A255380.

Vec((x-1)^2/(10*x-1)^2 + O(x^100)) \\ Colin Barker, Feb 27 2015

a(0) = 1, a(n) = (81n + 99) * 10^(n-2) for n >= 1.
G.f.: (x-1)^2/(10*x-1)^2. - Alois P. Heinz, Feb 27 2015
a(n) = 20*a(n-1) - 100*a(n-2) for n>2. - Colin Barker, Feb 27 2015

A365137 a(n) is the number of n-digit numbers that contain '22' in their decimal representation.

Original entry on oeis.org

0, 0, 1, 18, 261, 3411, 42048, 499131, 5770611, 65427678, 730784601, 8065910511, 88170256008, 956125498671, 10298661792111, 110293085617038, 1175325726682341, 12470569310694411, 131813055336390768, 1388552621823766611, 14583291094441416411, 152746593446386647198

Offset: 0

Felix Huber, Aug 23 2023

a(n) is also valid for '11', '33', '44', '55', '66', '77', '88' or '99' instead of '22'.

			a(2) = 1, the number 22 itself.
a(3) = 18, 10 numbers 22X plus 9 numbers X22 minus 1 number 222.
a(4) = 261, 100 numbers 22XX plus 90 numbers X22X plus 90 numbers XX22 minus 10 numbers 222X minus 9 numbers X222.

Felix Huber, Table of n, a(n) for n = 0..996
Armin Widmer, The number of Licence Plates that contain '22'
Index entries for linear recurrences with constant coefficients, signature (19,-81,-90).

Cf. A057092, A255372.

A365137 := proc(n) option remember; if n <= 1 then 0; elif n = 2 then 1; elif n = 3 then 18; else 19*procname(n - 1) - 81*procname(n - 2) - 90*procname(n - 3); end if; end proc; seq(A365137(n), n = 0 .. 21);

LinearRecurrence[{19, -81, -90}, {0, 0, 1, 18}, 22] (* Robert P. P. McKone, Aug 24 2023 *)

a(n) = 19*a(n - 1) - 81*a(n - 2) - 90*a(n - 3) with a(0) = a(1) = 0, a(2) = 1 and a(3) = 18 for n >= 4.
a(n) = 9*10^(n - 1) - A057092(n) + A057092(n - 2) with a(0) = a(1) = 0 for n >= 2.
a(0) = 0, a(n) = 9*10^(n - 1) - (p^(n + 1) - q^(n + 1))/(3*sqrt(13)) + (p^(n - 1) - q^(n - 1))/(3*sqrt(13)) with p = (9 + 3*sqrt(13))/2  and q = (9 - 3*sqrt(13))/2 for n >= 1.
G.f.: x^2*(1 - x)/((1 - 10*x)*(1 - 9*x - 9*x^2)).
a(n) = A255372(n) for n <= 5.

cp's OEIS Frontend

A255371 Number of strings of n decimal digits that contain at least one "0" digit that is not part of a string of two or more consecutive "0" digits.

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

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A255373 Number of strings of n decimal digits that contain at least one string of exactly 3 consecutive "0" digits.

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A255381 Number of strings of k+n decimal digits that contain one string of exactly k consecutive "0" digits, where k >= n.

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A365137 a(n) is the number of n-digit numbers that contain '22' in their decimal representation.

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