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

0, 1, 10, 141, 2548, 56197, 1463670, 43966297, 1496317768, 56904041481, 2391466059970, 110064342800101, 5505608606065020, 297412929070311181, 17255455494684113518, 1070135653599485349297, 70646208593060717167120, 4946304737167849687047697, 366097196759013937558696698

Offset: 1

Author

Gary Detlefs, Apr 26 2021

Keywords

• nonn

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 A343688. s(a,b,n) = a*A343688(n) + b*a(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.

a(n) mod n = 1 for even n and n-2 for odd n (empirical).

Examples

			a(4)=14*10+1, a(5)=18*141+10...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..366

Crossrefs

Cf. A340737, A340738, A343688.

Programs

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(0,1,n)) od

Mathematica

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 *)

Formula

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