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A338232 Number of ternary strings of length n that contain at least two 0's and at most two 1's.

%I A338232 #23 Jan 31 2025 13:50:14
%S A338232 0,0,1,7,33,121,378,1065,2803,7045,17148,40789,95373,220065,502414,
%T A338232 1136977,2553831,5699149,12645504,27914877,61337665,134213065,
%U A338232 292547346,635430937,1375724763,2969559381,6392110468,13723752805,29393671413,62813884465,133949278998,285078439329,605590372303
%N A338232 Number of ternary strings of length n that contain at least two 0's and at most two 1's.
%H A338232 Harvey P. Dale, <a href="/A338232/b338232.txt">Table of n, a(n) for n = 0..1000</a>
%H A338232 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (10,-42,96,-129,102,-44,8).
%F A338232 a(n) = 2^n + n*2^(n-1) + binomial(n,2)*2^(n-2) - 3*binomial(n,2) - 3*binomial(n,3) - 2*n - 1.
%F A338232 E.g.f.: exp(x)*(exp(x) - 1 - x)*(2 + 2*x + x^2)/2.
%F A338232 G.f.: x^2*(1-3*x+5*x^2-11*x^3+11*x^4)/((1-x)^4*(1-2*x)^3). - _Stefano Spezia_, Jan 31 2021
%e A338232 a(4) = 33 since the strings are composed of 0000, the 4 permutations of 0001, the 4 permutations of 0002, the 6 permutations of 0011, the 6 permutations of 0022, and the 12 permutations of 0012. Thus, the total number of strings is 1 + 4 + 4 + 6 + 6 + 12 = 33.
%t A338232 CoefficientList[Series[Exp[x](Exp[x]-1-x)(2+2x+x^2)/2,{x,0,32}],x]Table[i!,{i,0,32}] (* _Stefano Spezia_, Jan 31 2021 *)
%t A338232 LinearRecurrence[{10,-42,96,-129,102,-44,8},{0,0,1,7,33,121,378},40] (* _Harvey P. Dale_, Jan 31 2025 *)
%Y A338232 Cf. A338229, A338230.
%K A338232 nonn,easy
%O A338232 0,4
%A A338232 _Enrique Navarrete_, Jan 30 2021