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 A243019 #28 Aug 19 2025 18:41:50
%S A243019 1,1,5,7,33,51,233,379,1697,2851,12585,21627,94449,165075,714873,
%T A243019 1266027,5445441,9746883,41687369,75275227,320420753,582881971,
%U A243019 2471008281,4523575371,19108837601,35174066851
%N A243019 Expansion of -(2*x*sqrt(1-8*x^2)-2*x) / (16*x^3+sqrt(1-8*x^2)*(4*x^2+2*x-1)-8*x^2-2*x+1).
%C A243019 Number of ternary strings of length n that have the same number or more 0's than the combined number of 1's and 2's. For example, a(4) = 33 since the strings are (number of permutations in parentheses): 0000 (1), 0001 (4), 0002 (4), 0011 (6), 0022 (6), 0012 (12). - _Enrique Navarrete_, Aug 14 2025
%F A243019 a(n) = sum(i=0..floor(n/2), 2^i*binomial(n,i)).
%F A243019 G.f.: (x*C'(2*x^2))/(C(2*x^2)*(1-x*C(2*x^2))), where C(x) is g.f. of A000108.
%F A243019 a(n) ~ 2^(3*n/2) * (2+sqrt(2) + (-1)^n*(2-sqrt(2))) / sqrt(2*Pi*n). - _Vaclav Kotesovec_, May 29 2014
%F A243019 D-finite with recurrence: n^2*a(n) = (3*n^2-4)*a(n-1) + 4*(2*n^2 - 2*n - 1)*a(n-2) - 24*(n-2)*(n+1)*a(n-3). - _Vaclav Kotesovec_, May 29 2014
%F A243019 a(n) = Sum_{k=0..floor(n/2)} A013609(n,k). - _Enrique Navarrete_, Aug 14 2025
%t A243019 CoefficientList[Series[-(2*x*Sqrt[1-8*x^2]-2*x)/(16*x^3+Sqrt[1-8*x^2]*(4*x^2+2*x-1)-8*x^2-2*x+1),{x,0,20}],x] (* _Vaclav Kotesovec_, May 29 2014 *)
%o A243019 (Maxima)
%o A243019 a(n):=sum(2^(i)*binomial(n,i),i,0,floor((n)/2));
%Y A243019 Cf. A000108, A013609, A128418.
%K A243019 nonn,changed
%O A243019 0,3
%A A243019 _Vladimir Kruchinin_, May 29 2014