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

0,7

Links

Table of n, a(n) for n=0..29.

Index entries for linear recurrences with constant coefficients, signature (13,-72,222,-417,489,-350,140,-24).

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

Example

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.

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 *)

Crossrefs

Cf. A000453, A000478, A358341.

Sequence in context: A020868 A223461 A088943 * A097387 A057096 A246774

Adjacent sequences: A385309 A385310 A385311 * A385313 A385314 A385315

Keyword

nonn,easy

Author

Enrique Navarrete, Jun 25 2025

Status

approved