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A343688 a(1)=1, a(2)=0, a(n) = (4*n-2)*a(n-1) + a(n-2), n > 2.

%I A343688 #30 Jun 03 2021 09:10:40
%S A343688 1,0,1,14,253,5580,145333,4365570,148574713,5650204664,237457170601,
%T A343688 10928680052310,546671459786101,29531187508501764,1713355546952888413,
%U A343688 106257575098587583370,7014713312053733390833,491136189418859924941680,36351092730307688179075153
%N A343688 a(1)=1, a(2)=0, a(n) = (4*n-2)*a(n-1) + a(n-2), n > 2.
%C A343688 This sequence is one of the two "basis" sequences for sequences having the form s(a,b,1)=a, s(a,b,2)=b, s(n) = (4*n-2)*s(a,b,n-1) + s(a,b,n-2), the second being A343689. s(a,b,n) = a*a(n) + b*A343689(n).
%C A343688 Of specific interest is s(3,19,n) and s(1,7,n) which produce the odd terms of A340737 and A340738 respectively and whose quotient converges to e.
%C A343688 It is of interest to note that a(n)*A343689(n+1) - a(n+1)*A343689(n) = (-1)^(n+1), a(n)*A343689(n+2) - a(n+2)*A343689(n) = (4*n+6)*(-1)^(n+1) and a(n)*A343689(n+3) - a(n+3)*A343689(n) =((4*n+8)^2-3)* (-1)^(n+1)
%C A343688 a(n) mod n = n-6 for even n > 4 and 13 for odd n > 13 (empirical).
%F A343688 a(1)=1, a(2)=0, a(n) = (4*n-2)*a(n-1) + a(n-2), n > 2.
%e A343688 a(4)=14*1+0, a(5)=18*14+1, ...
%p A343688 e := proc(a, b, n) option remember; if n = 1 then a; else if n = 2 then b; else (4*n - 2)*e(a, b, n - 1) + e(a, b, n - 2); end if; end if; end proc;
%p A343688 for n from 1 to 20 do print(e(1,0,n)) od
%t A343688 a[1]=1;a[2]=0;a[n_]:=a[n]=(4n-2)a[n-1]+a[n-2];Array[a,20] (* _Giorgos Kalogeropoulos_, Apr 27 2021 *)
%Y A343688 Cf. A340737, A340738, A343689.
%K A343688 nonn
%O A343688 1,4
%A A343688 _Gary Detlefs_, Apr 26 2021