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A134771 A134770 interleaved with threes.

%I A134771 #20 Oct 14 2023 06:56:33
%S A134771 1,3,5,3,21,3,77,3,277,3,1005,3,3693,3,13725,3,51477,3,194477,3,
%T A134771 739021,3,2821725,3,10816621,3,41602397,3,160466397,3,620470077,3,
%U A134771 2404321557,3,9334424877,3,36300541197,3,141381055197,3,551386115277,3,2153031497757,3
%N A134771 A134770 interleaved with threes.
%C A134771 Previous name was: A007318^(-2) * A134770.
%C A134771 Second inverse binomial transform of A134770.
%C A134771 A134770 interleaved with threes.
%H A134771 G. C. Greubel, <a href="/A134771/b134771.txt">Table of n, a(n) for n = 0..1000</a>
%F A134771 From _G. C. Greubel_, Oct 13 2023: (Start)
%F A134771 a(n) = 2*(1 + (-1)^n)*binomial(n, n/2) - 3*(-1)^n.
%F A134771 G.f.: 2/sqrt(1-4*x^2) - 3/(1+x).
%F A134771 E.g.f.: 4*BesselI(0, 2*x) - 3*exp(-x). (End)
%e A134771 First few terms of the sequence are (1, 3, 5, 3, 21, 3, 77, ...), since A134770 = (1, 3, 5, 21, 77, ...).
%t A134771 Table[If[OddQ[n], 3, 4*Binomial[n,n/2] -3], {n,0,50}] (* _G. C. Greubel_, Oct 13 2023 *)
%o A134771 (Magma)
%o A134771 A134771:= func< n | (n mod 2) eq 1 select 3 else 2*(n+2)*Catalan(Floor(n/2))-3 >;
%o A134771 [A134771(n): n in [0..50]]; // _G. C. Greubel_, Oct 13 2023
%o A134771 (SageMath)
%o A134771 def A134771(n): return 4*((n+1)%2)*binomial(n, n//2) - 3*(-1)^n
%o A134771 [A134771(n) for n in range(41)] # _G. C. Greubel_, Oct 13 2023
%Y A134771 Cf. A000984, A010701, A134770.
%K A134771 nonn,easy,less
%O A134771 0,2
%A A134771 _Gary W. Adamson_, Nov 10 2007
%E A134771 Name changed by _G. C. Greubel_, Oct 13 2023