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

1, 0, 1, 14, 253, 5580, 145333, 4365570, 148574713, 5650204664, 237457170601, 10928680052310, 546671459786101, 29531187508501764, 1713355546952888413, 106257575098587583370, 7014713312053733390833, 491136189418859924941680, 36351092730307688179075153

Offset

1,4

Comments

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).

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.

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)

a(n) mod n = n-6 for even n > 4 and 13 for odd n > 13 (empirical).

Links

Table of n, a(n) for n=1..19.

Formula

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

Example

a(4)=14*1+0, a(5)=18*14+1, ...

Maple

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;
for n from 1 to 20 do print(e(1, 0, n)) od

Mathematica

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 *)

Crossrefs

Cf. A340737, A340738, A343689.

Sequence in context: A051622 A113377 A318596 * A159516 A005611 A304446

Adjacent sequences: A343685 A343686 A343687 * A343689 A343690 A343691

Keyword

nonn

Author

Gary Detlefs, Apr 26 2021

Status

approved