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A378676 a(n) = J(n) * J(n+2) where J(n) = Jacobsthal(n) = A001045(n).

%I A378676 #11 Dec 08 2024 17:13:13
%S A378676 0,3,5,33,105,473,1785,7353,28985,116793,465465,1865273,7454265,
%T A378676 29830713,119295545,477236793,1908837945,7635570233,30541844025,
%U A378676 122168249913,488671252025,1954688503353,7818747022905,31275002072633,125099980328505,500399977238073,2001599797104185,8006399412112953,32025597201059385
%N A378676 a(n) = J(n) * J(n+2) where J(n) = Jacobsthal(n) = A001045(n).
%H A378676 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-8).
%F A378676 a(n) = (2^n - (-1)^n) * (2^(n+2) - (-1)^n) / 9 = (4 * 4^n - 5 * (-2)^n + 1) / 9.
%F A378676 G.f.: x * (3 - 4*x) / ((1-x) * (1+2*x) * (1-4*x)).
%F A378676 a(n) = 3 * a(n-1) + 6 * a(n-2) - 8 * a(n-3) for n > 2 with initial values a(0) = 0, a(1) = 3, and a(2) = 5.
%F A378676 Sum_{k=1..n-1} 2^(k-1) / a(k) = 1 - 2^(n-1) / A084175(n) for n > 0.
%F A378676 Sum_{k>0} 2^(k-1) / a(k) = 1.
%F A378676 E.g.f.: exp(x)*(1 - cosh(3*x) + 9*sinh(3*x))/9. - _Stefano Spezia_, Dec 06 2024
%t A378676 a[n_] := (4^(n+1) - 5*(-2)^n + 1)/9; Array[a, 30, 0] (* _Amiram Eldar_, Dec 06 2024 *)
%o A378676 (PARI) a(n)=(4^(n+1)-5*(-2)^n+1)/9
%Y A378676 Cf. A001045, A084175.
%K A378676 nonn,easy
%O A378676 0,2
%A A378676 _Werner Schulte_, Dec 03 2024