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 A307018 #38 Dec 11 2024 15:53:51
%S A307018 0,0,0,1,0,1,2,1,2,4,2,4,6,4,6,9,6,9,12,9,12,16,12,16,20,16,20,25,20,
%T A307018 25,30,25,30,36,30,36,42,36,42,49,42,49,56,49,56,64,56,64,72,64,72,81,
%U A307018 72,81,90,81,90,100,90,100,110,100,110,121,110,121,132
%N A307018 Total number of parts of size 3 in the partitions of n into parts of size 2 and 3.
%H A307018 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2,0,-2,-1,0,1).
%F A307018 a(n+2) = A321202(n) - A114209(n+1).
%F A307018 a(3n+1) = A002620(n+2).
%F A307018 a(3n+2) = A002620(n+1).
%F A307018 a(3n+3) = A002620(n+2).
%F A307018 G.f.: x^3/((1+x)*(1+x+x^2)^2*(1-x)^3). - _Alois P. Heinz_, Mar 19 2019
%F A307018 a(n) = a(n-2) + 2*a(n-3) - 2*a(n-5) - a(n-6) + a(n-8). - _G. C. Greubel_, Apr 03 2019
%F A307018 a(n) = (6*n*(2 + n) + 8*(4 + 3*n)*cos(2*n*Pi/3) - 8*sqrt(3)*n*sin(2*n*Pi/3) - 5 - 27*(-1)^n)/216. - _Stefano Spezia_, Apr 21 2022
%F A307018 From _Ridouane Oudra_, Nov 24 2024: (Start)
%F A307018 a(n) = (7*n/2 - 7*n^2/2 - 9*floor(n/2) + (6*n+4)*floor(2*n/3) + 4*floor(n/3))/18.
%F A307018 a(n) = A008133(n) - A069905(n-1).
%F A307018 a(n) = A002620(A008611(n)). (End)
%t A307018 LinearRecurrence[{0,1,2,0,-2,-1,0,1}, {0,0,0,1,0,1,2,1}, 80] (* _G. C. Greubel_, Apr 03 2019 *)
%t A307018 Table[(6n(2+n)-5-27(-1)^n+8(4+3n)Cos[2n Pi/3]-8Sqrt[3]n Sin[2n Pi/3])/216,{n,0,66}] (* _Stefano Spezia_, Apr 21 2022 *)
%o A307018 (PARI) my(x='x+O('x^80)); concat([0,0,0], Vec(x^3/((1-x^2)*(1-x^3)^2))) \\ _G. C. Greubel_, Apr 03 2019
%o A307018 (Magma) R<x>:=PowerSeriesRing(Integers(), 80); [0,0,0] cat Coefficients(R!( x^3/((1-x^2)*(1-x^3)^2) )); // _G. C. Greubel_, Apr 03 2019
%o A307018 (Sage) (x^3/((1-x^2)*(1-x^3)^2)).series(x, 80).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 03 2019
%o A307018 (GAP) a:=[0,0,0,1,0,1,2,1];; for n in [9..80] do a[n]:=a[n-2]+2*a[n-3] -2*a[n-5]-a[n-6]+a[n-8]; od; a; # _G. C. Greubel_, Apr 03 2019
%Y A307018 Cf. A000041, A002620, A103221, A114209, A321202.
%Y A307018 Cf. A008133, A069905, A008611.
%K A307018 nonn,easy
%O A307018 0,7
%A A307018 _Andrew Ivashenko_, Mar 19 2019
%E A307018 More terms from _Alois P. Heinz_, Mar 19 2019