OFFSET
1,2
COMMENTS
a(n) is the number of n-digit numbers in A031955.
LINKS
Stefano Spezia, Table of n, a(n) for n = 1..3300
Puzzle Critic, Twitter post about the case n = 5, Jul 16 2020.
Index entries for linear recurrences with constant coefficients, signature (3,-2).
FORMULA
O.g.f.: 81*x^2/(1 - 3*x + 2*x^2).
E.g.f.: 81*(exp(x) - 1)^2/2.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 2.
a(n) = 81*(2^(n-1) - 1).
a(n) = 81*A000225(n-1).
EXAMPLE
a(1) = 0 since the positive integers must have at least two digits;
a(2) = 81 since #[99] - #[9] - #(11*[9]) = 99 - 9 - 9 = 81;
a(3) = 243 since #[999] - #[99] - #(111*[9]) - #{xyz in N | x,y,z are three different digits with x != 0} = 999 - 99 - 9 - 9*9*8 = 243;
...
MATHEMATICA
LinearRecurrence[{3, -2}, {0, 81}, 31]
PROG
(PARI) concat([0], Vec(81*x^2/(1-3*x+2*x^2)+O(x^31)))
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
Stefano Spezia, Jul 18 2020
EXTENSIONS
a(0) removed by Stefano Spezia, Sep 23 2020
STATUS
approved