|
%I #32 Jun 07 2018 11:09:56
%S 2,4,14,32,82,196,478,1152,2786,6724,16238,39200,94642,228484,551614,
%T 1331712,3215042,7761796,18738638,45239072,109216786,263672644,
%U 636562078,1536796800,3710155682,8957108164,21624372014,52205852192
%N a(n) = -(u^n-1)*(v^n-1) with u = 1+sqrt(2), v = 1-sqrt(2).
%H Seiichi Manyama, <a href="/A129744/b129744.txt">Table of n, a(n) for n = 1..1000</a>
%H G. Everest et al., <a href="http://www.jstor.org/stable/27642221">Primes generated by recurrence sequences</a>, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).
%F a(2n) = A002203(2n)-2. a(2n+1) = A002203(2n+1). - _R. J. Mathar_, corrected Dec 05 2007.
%F G.f.: 2*x*(1+x^2)/((x^2+2*x-1)*(-1+x)*(1+x)).
%F From _Peter Bala_, Mar 19 2015: (Start)
%F a(n) = -det(I - M^n) where I is the 2X2 identity matrix and M = [2, 1; 1, 0]. Cf. A001350.
%F a(n) = 2*A113224(n-1).
%F This is divisibility sequence, that is, if n | m then a(n) | a(m).
%F exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*Sum_{n >= 1} Pell(n) *x^n. (End)
%F 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
%p 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
%t Table[Simplify[ -((1 + Sqrt[2])^n - 1)*((1 - Sqrt[2])^n - 1)], {n, 1, 30}] (* _Stefan Steinerberger_, May 15 2007 *)
%o (PARI) w = quadgen(8); vector(30, n, -((1+w)^n-1)*((1-w)^n-1)) \\ _Michel Marcus_, Mar 21 2015
%o (PARI) Vec(2*x*(1+x^2)/((x^2+2*x-1)*(-1+x)*(1+x))+O(x^99)) \\ _Charles R Greathouse IV_, Nov 13 2015
%Y Cf. A002003, A000129, A001350, A113224.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, May 13 2007
%E More terms from _Emeric Deutsch_ and _Stefan Steinerberger_, May 13 2007
|
