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 A100068 #30 Jan 15 2025 06:36:33
%S A100068 1,3,11,36,123,408,1370,4560,15235,50760,169326,564336,1881582,
%T A100068 6271632,20907156,69689376,232304355,774343560,2581169510,8603882160,
%U A100068 28679699578,95598937008,318663476076,1062211351776,3540705857998,11802351958608,39341178395660,131137257852000
%N A100068 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*3^(n-2*k).
%C A100068 An inverse Chebyshev transform of x/(1-3*x), where the Chebyshev transform of g(x) is ((1-x^2)/(1+x^2))*g(x/(1+x^2)) and the inverse transform maps a g.f. A(x) to (1/sqrt(1-4*x^2))*A(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108. In general, Sum_{k=0..floor(n/2)} binomial(n,k) * r^(n-2*k) has g.f. 2*x/(sqrt(1-4*x^2)*(r*sqrt(1-4*x^2) + 2*x - r)). - corrected by _Vaclav Kotesovec_, Dec 06 2012
%C A100068 Generally (for r>1), a(n) ~ (r + 1/r)^n. - _Vaclav Kotesovec_, Dec 06 2012
%H A100068 Vincenzo Librandi, <a href="/A100068/b100068.txt">Table of n, a(n) for n = 0..1000</a>
%F A100068 G.f.: 2*x/(sqrt(1-4*x^2)*(3*sqrt(1-4*x^2)+2*x-3)).
%F A100068 a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*3^(n-k).
%F A100068 a(n) = Sum_{k=0..n} binomial(n, (n-k)/2)*(1+(-1)^(n-k))*3^k/2.
%F A100068 D-finite with recurrence 9*n*a(n) +12*(-3*n+1)*a(n-1) +4*(-4*n-1)*a(n-2) +48*(3*n-4)*a(n-3) +80*(-n+3)*a(n-4)=0. - _R. J. Mathar_, Nov 22 2012
%F A100068 a(n) ~ 10^n/3^n. - _Vaclav Kotesovec_, Dec 06 2012
%t A100068 CoefficientList[Series[2*x/(Sqrt[1-4*x^2]*(3*Sqrt[1-4*x^2] + 2*x-3)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Dec 06 2012 *)
%o A100068 (PARI) my(x='x+O('x^66)); Vec(2*x/(sqrt(1-4*x^2)*(3*sqrt(1-4*x^2)+2*x-3))) \\ _Joerg Arndt_, May 12 2013
%o A100068 (Magma) m:=3; [(&+[Binomial(n,k)*m^(n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // _G. C. Greubel_, Jun 08 2022
%o A100068 (SageMath) m=3; [sum(binomial(n,k)*m^(n-2*k) for k in (0..n//2)) for n in (0..40)] # _G. C. Greubel_, Jun 08 2022
%Y A100068 Cf. A027306, A100067, A100069.
%K A100068 easy,nonn
%O A100068 0,2
%A A100068 _Paul Barry_, Nov 02 2004