|
|
A129744
|
|
a(n) = -(u^n-1)*(v^n-1) with u = 1+sqrt(2), v = 1-sqrt(2).
|
|
5
|
|
|
2, 4, 14, 32, 82, 196, 478, 1152, 2786, 6724, 16238, 39200, 94642, 228484, 551614, 1331712, 3215042, 7761796, 18738638, 45239072, 109216786, 263672644, 636562078, 1536796800, 3710155682, 8957108164, 21624372014, 52205852192
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
LINKS
|
|
|
FORMULA
|
G.f.: 2*x*(1+x^2)/((x^2+2*x-1)*(-1+x)*(1+x)).
a(n) = -det(I - M^n) where I is the 2X2 identity matrix and M = [2, 1; 1, 0]. Cf. A001350.
This is divisibility sequence, that is, if n | m then a(n) | a(m).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*Sum_{n >= 1} Pell(n) *x^n. (End)
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n > 4. - Seiichi Manyama, Jun 07 2018
|
|
MAPLE
|
u:=1+sqrt(2): v:=1-sqrt(2): a:=n->expand(-(u^n-1)*(v^n-1)): seq(a(n), n=1..33); # Emeric Deutsch, May 13 2007
|
|
MATHEMATICA
|
Table[Simplify[ -((1 + Sqrt[2])^n - 1)*((1 - Sqrt[2])^n - 1)], {n, 1, 30}] (* Stefan Steinerberger, May 15 2007 *)
|
|
PROG
|
(PARI) w = quadgen(8); vector(30, n, -((1+w)^n-1)*((1-w)^n-1)) \\ Michel Marcus, Mar 21 2015
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|