A180599 -id:A180599 - cp's OEIS frontend

A337313 a(n) is the number of n-digit positive integers with exactly three distinct base 10 digits.

0, 0, 648, 3888, 16200, 58320, 195048, 625968, 1960200, 6045840, 18468648, 56068848, 169533000, 511252560, 1539065448, 4627812528, 13904670600, 41756478480, 125354369448, 376232977008, 1129038669000, 3387795483600, 10164745404648, 30496954122288, 91496298184200

Offset: 1

Stefano Spezia, Aug 22 2020

a(n) is the number of n-digit numbers in A031962.

			a(1) = a(2) = 0 since the positive integers must have at least three digits;
a(3) = #{xyz in N | x,y,z are three different digits with x != 0} = 9*9*8 = 648;
a(4) = 3888 since #[9999] - #[999] - #(1111*[9]) - A335843(4) - #{xywz in N | x,y,w,z are four different digits with x != 0} = 9999 - 999 - 9 - 567 - 9*9*8*7 = 3888;
...

Index entries for linear recurrences with constant coefficients, signature (6,-11,6).
Index entries for sequences related to digits.

Cf. A000225, A000244, A000392, A031962, A048993, A335843, A337127.

Cf. A055642, A180599, A235155, A235691, A235718.

LinearRecurrence[{6,-11,6},{0,0,648},26]

concat([0,0],Vec(648*x^3/(1-6*x+11*x^2-6*x^3)+O(x^26)))

O.g.f.: 648*x^3/(1 - 6*x + 11*x^2 - 6*x^3).
E.g.f.: 108*(exp(x) - 1)^3.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
a(n) = 648*S2(n, 3) where S2(n, 3) = A000392(n).
a(n) = 324*(3^(n-1) - 2^n + 1).
a(n) ~ 108 * 3^n.
a(n) = 324*(A000244(n-1) - A000225(n)).
a(n) = A337127(n, 3).

A291092 1 followed by infinitely many 9's.

Original entry on oeis.org

1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 1

N. J. A. Sloane, Aug 19 2017

The digital root of 9^(n-1). - Cino Hilliard, Dec 31 2004

With interpolated zeros (1,0,9,0,9,0,9,0,...) this is the number of hours between times when the hands of a two-handed clock cross. - Halfdan Skjerning, Aug 18 2017

Brady Haran and Cliff Stoll, When do clock hands overlap?, Numberphile video (2017)
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A178181, A010734, A180599, A100403, A100401.

PadRight[{1},100,9] (* Paolo Xausa, Oct 16 2023 *)

G.f.: x*(1 + 8*x)/(1 - x). - Chai Wah Wu, Aug 19 2017

E.g.f.: 9*(exp(x) - 1) - 8*x. - Stefano Spezia, Oct 16 2023

A337314 a(n) is the number of n-digit positive integers with exactly four distinct base 10 digits.

Original entry on oeis.org

0, 0, 0, 4536, 45360, 294840, 1587600, 7715736, 35244720, 154700280, 661122000, 2773768536, 11487556080, 47136955320, 192126589200, 779279814936, 3149513947440, 12695388483960, 51073849285200, 205172877726936, 823325141746800, 3301203837670200, 13228529919066000

Offset: 1

Stefano Spezia, Sep 26 2020

a(n) is the number of n-digit numbers in A031969.

			a(1) = a(2) = a(3) = 0 since the positive integers must have at least four digits;
a(4) = #{wxyz in N | w,x,y,z are four different digits with w != 0} = A073531(4) = 4536;
a(5) = 45360 since #[99999] - #[9999] - #(11111*[9]) - A335843(5) - A337313(5) - #{vwxyz in N | v,w,x,y,z are five different digits with v != 0} = 99999 - 9999 - 9 - 1215 - 16200 - 9*9*8*7*6 = 45360;
...

Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).
Index entries for sequences related to digits.

Cf. A000079, A000244, A000302, A000453, A031969, A073531, A335843, A337313, A337127.

Cf. A055642, A180599, A235155, A235691, A235718.

LinearRecurrence[{10,-35,50,-24},{0,0,0,4536},23]

concat([0,0,0],Vec(4536*x^4/(1-10*x+35*x^2-50*x^3+24*x^4)+O(x^24)))

O.g.f.: 4536*x^4/(1 - 10*x + 35*x^2 - 50*x^3 + 24*x^4).
E.g.f.: 189*(exp(x) - 1)^4.
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4) for n > 4.
a(n) = 4536*S2(n, 4) where S2(n, 4) = A000453(n).
a(n) = 189*(4^n - 4*3^n + 3*2^(n+1) - 4).
a(n) ~ 189 * 4^n.
a(n) = 189*(A000302(n) - 4*A000244(n) + 3*A000079(n+1) - 4).
a(n) = A337127(n, 4).

cp's OEIS Frontend

A337313 a(n) is the number of n-digit positive integers with exactly three distinct base 10 digits.

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A291092 1 followed by infinitely many 9's.

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A337314 a(n) is the number of n-digit positive integers with exactly four distinct base 10 digits.

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Author

Keywords

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Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula