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A343689 a(1)=0, a(2)=1, a(n) = (4*n-2)*a(n-1) + a(n-2), n > 2.

%I A343689 #20 Dec 17 2021 13:06:31
%S A343689 0,1,10,141,2548,56197,1463670,43966297,1496317768,56904041481,
%T A343689 2391466059970,110064342800101,5505608606065020,297412929070311181,
%U A343689 17255455494684113518,1070135653599485349297,70646208593060717167120,4946304737167849687047697,366097196759013937558696698
%N A343689 a(1)=0, a(2)=1, a(n) = (4*n-2)*a(n-1) + a(n-2), n > 2.
%C A343689 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 A343688. s(a,b,n) = a*A343688(n) + b*a(n).
%C A343689 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 A343689 a(n) mod n = 1 for even n and n-2 for odd n (empirical).
%H A343689 Harvey P. Dale, <a href="/A343689/b343689.txt">Table of n, a(n) for n = 1..366</a>
%F A343689 a(1)=0, a(1)=1, a(n) = (4*n-2)*a(n-1) + a(n-2).
%e A343689 a(4)=14*10+1, a(5)=18*141+10...
%p A343689 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 A343689 for n from 1 to 20 do print(e(0,1,n)) od
%t A343689 RecurrenceTable[{a[1]==0,a[2]==1,a[n]==(4n-2)a[n-1]+a[n-2]},a,{n,20}] (* _Harvey P. Dale_, Dec 17 2021 *)
%Y A343689 Cf. A340737, A340738, A343688.
%K A343689 nonn
%O A343689 1,3
%A A343689 _Gary Detlefs_, Apr 26 2021