OFFSET
1,2
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
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 linear recurrences with constant coefficients, signature (19,-90).
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
EXAMPLE
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.
MAPLE
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 *)
PROG
(PARI) a(n)=9*10^n-8*9^n \\ M. F. Hasler, May 04 2015
(Magma) [9*10^(n-1) - 8*9^(n-1): n in [1..30]]; // Vincenzo Librandi, May 04 2015
CROSSREFS
KEYWORD
base,easy,nonn
AUTHOR
Marc LeBrun, Oct 23 2003
STATUS
approved