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a(n) is the number of n-digit positive integers with exactly three distinct base 10 digits.
3

%I #23 Sep 03 2020 12:15:15

%S 0,0,648,3888,16200,58320,195048,625968,1960200,6045840,18468648,

%T 56068848,169533000,511252560,1539065448,4627812528,13904670600,

%U 41756478480,125354369448,376232977008,1129038669000,3387795483600,10164745404648,30496954122288,91496298184200

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

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

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).

%H <a href="/index/Di#digits">Index entries for sequences related to digits</a>.

%F O.g.f.: 648*x^3/(1 - 6*x + 11*x^2 - 6*x^3).

%F E.g.f.: 108*(exp(x) - 1)^3.

%F a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.

%F a(n) = 648*S2(n, 3) where S2(n, 3) = A000392(n).

%F a(n) = 324*(3^(n-1) - 2^n + 1).

%F a(n) ~ 108 * 3^n.

%F a(n) = 324*(A000244(n-1) - A000225(n)).

%F a(n) = A337127(n, 3).

%e a(1) = a(2) = 0 since the positive integers must have at least three digits;

%e a(3) = #{xyz in N | x,y,z are three different digits with x != 0} = 9*9*8 = 648;

%e 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;

%e ...

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

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

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

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

%K nonn,easy,base

%O 1,3

%A _Stefano Spezia_, Aug 22 2020