A385312 a(n) is the number of ternary strings of length n with at least one 0, at least two 1's and at least three 2's.
0, 0, 0, 0, 0, 0, 60, 455, 2268, 9366, 34800, 121077, 403392, 1304732, 4133220, 12900771, 39837684, 122064930, 371891592, 1128317489, 3412864056, 10299925992, 31033986588, 93394501983, 280818931020, 843832511150, 2534467085280, 7609793357805, 22843103816688, 68558705110836
Offset: 0
Examples
a(6) = 60 since the strings are the 60 permutations of 011222. a(7) = 455 since the strings are the 210 permutations of 0011222, the 140 permutations of 0111222 and the 105 permutations of 0112222.
Links
- Index entries for linear recurrences with constant coefficients, signature (13,-72,222,-417,489,-350,140,-24).
Programs
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Mathematica
LinearRecurrence[{13, -72, 222, -417, 489, -350, 140, -24}, {0, 0, 0, 0, 0, 0, 60, 455, 2268, 9366, 34800, 121077}, 30] (* Amiram Eldar, Jun 28 2025 *)
Formula
a(n) = 3^n - 2^(n-2)*(binomial(n,2) + 4*n + 12) + 3*binomial(n,3) + 4*binomial(n,2) + 4*n + 3 for n>=4.
E.g.f.: (exp(x) - x^2/2 - x - 1)*(exp(x) - x - 1)*(exp(x) - 1).
G.f.: x^6*(60 - 325*x +673*x^2 - 678*x^3 + 348*x^4 - 72*x^5)/((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)). - Stefano Spezia, Jun 25 2025