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 A327319 #52 May 23 2025 18:39:25
%S A327319 0,0,1,3,12,40,105,231,448,792,1305,2035,3036,4368,6097,8295,11040,
%T A327319 14416,18513,23427,29260,36120,44121,53383,64032,76200,90025,105651,
%U A327319 123228,142912,164865,189255,216256,246048,278817,314755,354060,396936
%N A327319 a(n) = binomial(n, 2) + 6*binomial(n, 4).
%C A327319 a(n) is the number of ternary strings of length n that have exactly two 2's, zero or two 1's, and have no restriction on the number of 0's. For example, a(6)=105 since the strings are the 90 permutations of 221100 and the 15 permutations of 220000. - _Enrique Navarrete_, May 19 2025
%H A327319 Colin Barker, <a href="/A327319/b327319.txt">Table of n, a(n) for n = 0..1000</a>
%H A327319 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A327319 From _Colin Barker_, Sep 21 2019: (Start)
%F A327319 G.f.: x^2*(1 - 2*x + 7*x^2) / (1 - x)^5.
%F A327319 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
%F A327319 a(n) = (n*(-8 + 13*n - 6*n^2 + n^3)) / 4. (End)
%F A327319 E.g.f.: (1/4)*exp(x)*x^2*(2 + x^2). - _Stefano Spezia_, Sep 21 2019
%e A327319 a(5) = binomial(5, 2) + 6*binomial(5, 4) = 10 + 6*5 = 40.
%t A327319 Table[Binomial[n, 2] + 6Binomial[n, 4], {n, 0, 39}] (* _Alonso del Arte_, Sep 18 2019 *)
%t A327319 LinearRecurrence[{5,-10,10,-5,1},{0,0,1,3,12},40] (* _Harvey P. Dale_, Dec 10 2022 *)
%o A327319 (PARI) a(n) = {binomial(n, 2) + 6 * binomial(n, 4)} \\ _Andrew Howroyd_, Sep 20 2019
%o A327319 (PARI) concat([0,0], Vec(x^2*(1 - 2*x + 7*x^2) / (1 - x)^5 + O(x^40))) \\ _Colin Barker_, Sep 25 2019
%K A327319 nonn,easy
%O A327319 0,4
%A A327319 _Aaron Kemats_, Sep 17 2019