A088924 Number of "9ish numbers" with n digits.

1, 18, 252, 3168, 37512, 427608, 4748472, 51736248, 555626232, 5900636088, 62105724792, 648951523128, 6740563708152, 69665073373368, 716985660360312, 7352870943242808, 75175838489185272, 766582546402667448

Offset: 1

Author

Marc LeBrun, Oct 23 2003

Keywords

• base
• easy
• nonn

Comments

First difference of A016189. ("9" can be replaced by any other nonzero digit, however only the 9ish numbers are closed under lunar multiplication.)

See A257285 - A257289 for first differences of 5^n-4^n, ..., 9^n-8^n. These also give the number of n-digit numbers whose largest digit is 5, 6, 7, 8, respectively. - M. F. Hasler, May 04 2015

Examples

			a(2) = 18 because 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98 and 99 are the eighteen two-digit 9ish numbers.
		

Links

• Stefano Spezia, Table of n, a(n) for n = 1..950
• D. Applegate, C program for lunar arithmetic and number theory [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
• D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
• Index entries for sequences related to dismal (or lunar) arithmetic
• Index entries for linear recurrences with constant coefficients, signature (19,-90).

Crossrefs

Cf. A016189, A011539, A257285, A257286, A257287, A257288, A257289.

Programs

Magma

[9*10^(n-1) - 8*9^(n-1): n in [1..30]]; // Vincenzo Librandi, May 04 2015

Maple

A088924:=n->9*10^(n-1) - 8*9^(n-1); seq(A088924(n), n=1..30); # Wesley Ivan Hurt, May 15 2014

Mathematica

Series[(x (1 - x))/(1 - 19 x + 90 x^2), {x, 0, 10}] (* Bobby Milazzo, May 02 2014 *)
Table[9*10^(n - 1) - 8*9^(n - 1), {n, 30}] (* Wesley Ivan Hurt, May 15 2014 *)

PARI

a(n)=9*10^n-8*9^n \\ M. F. Hasler, May 04 2015

Python

def A088924(n): return 9*10**(n-1)-8*9**(n-1) # Chai Wah Wu, Jan 27 2025

Formula

a(n) = 9*10^(n-1) - 8*9^(n-1).

G.f.: x*(1 - x)/(1 - 19*x + 90*x^2). - Bobby Milazzo, May 02 2014

a(n) = 19*a(n-1) - 90*a(n-2). - Vincenzo Librandi, May 04 2015

E.g.f.: (81*exp(10*x) - 80*exp(9*x) - 1)/90. - Stefano Spezia, Nov 16 2023