This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309000 #26 Sep 08 2022 08:46:21
%S A309000 3,22,105,416,1491,5034,16365,51892,161799,498686,1524705,4635528,
%T A309000 14037627,42391378,127763925,384536924,1156232175,3474201510,
%U A309000 10434138825,31326533680,94029932643,282194655482,846802070205,2540859195396,7623517110231,22872497487694
%N A309000 Number of strings of length n from a 3-symbol alphabet (A,B,C, say) containing at least one "A" and at least two "B"s.
%C A309000 This sequence can be thought of as the number of ways of rolling n 3-sided dice (with sides "A", "B", and "C") and obtaining at least one A and at least two B's.
%C A309000 The general formula is readily proved true by counting arguments.
%H A309000 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,-31,51,-40,12).
%F A309000 a(n) = 3^n - 2^(n+1) - n*2^(n-1) + n + 1.
%F A309000 G.f.: x^3*(-3 + 5*x)/((-1 + 3*x)*(1 - 3*x + 2*x^2)^2). - _Michael De Vlieger_, Jul 04 2019.
%F A309000 a(n) = 9*a(n-1) - 31*a(n-2) + 51*a(n-3) - 40*a(n-4) + 12*a(n-5) for n > 7. - _Stefano Spezia_, Jul 05 2019
%e A309000 Suppose three-sided dice each have sides labeled A,B,C.
%e A309000 If there are three dice, then ABB, BAB, and BBA are the three strings resulting from rolling the dice satisfying the property of at least one A and at least two B's, hence a(3)=3 [Note a(0)=a(1)=a(2)=0].
%e A309000 If there are four such dice, there are 22 such permutations, hence a(4)=22: AABB, ABAB, ABBA, ABBB, ABBC, ABCB, ACBB, BAAB, BABA, BABB, BABC, BACB, BBAA, BBAB, BBAC, BBBA, BBCA, BCAB, BCBA, CABB, CBAB, CBBA.
%t A309000 Array[3^# - 2^(# + 1) - # 2^(# - 1) + # + 1 &, 27, 3] (* or *)
%t A309000 CoefficientList[Series[(-3 + 5 x)/((-1 + 3 x) (1 - 3 x + 2 x^2)^2), {x, 0, 26}], x] (* _Michael De Vlieger_, Jul 04 2019 *)
%o A309000 (Python) [3**n-2**(n+1)-n*2**(n-1)+n+1 for n in range(3,20)]
%o A309000 (Magma) [3^n-2^(n+1)-n*2^(n-1)+n+1: n in [3..40]]; // _Vincenzo Librandi_, Jul 05 2019
%Y A309000 Cf. A269914, A269915, A269916, A269917, A186244, A186314.
%K A309000 nonn,easy
%O A309000 3,1
%A A309000 _Adam Vellender_, Jul 04 2019