A144844 a(n) = ((2 + sqrt(2))^n - (2 - sqrt(2))^n)^2/8.

0, 1, 16, 196, 2304, 26896, 313600, 3655744, 42614784, 496754944, 5790601216, 67500196864, 786839961600, 9172078759936, 106917585289216, 1246322708463616, 14528202160472064, 169353135091941376, 1974124812461670400, 23012085209172803584

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..950

Index entries for linear recurrences with constant coefficients, signature (14,-28,8).

Formula

From R. J. Mathar, Sep 24 2008: (Start)

G.f.: x*(1+2*x)/((1-2*x)*(1-12*x+4*x^2)).

a(n) = 2^(n-2)*(A001109(n+1) - 3*A001109(n) - 1) = 2^(n-1)*A001108(n). (End)

a(n) = 14*a(n-1) - 28*a(n-2) + 8*a(n-3) for n > 2; a(0) = 0, a(1) = 1; a(2) = 16. - Klaus Brockhaus, Jul 15 2009

a(n) = A007070(n)^2 = (((sqrt(2)+1)^n - (sqrt(2)-1)^n)) / 2) ^ 2. - Franklin T. Adams-Watters, Aug 06 2009

a(n) = 2^(n-3)*(Q(2*n) - 2), where Q(m) are the Pell-Lucas numbers (A002203). - G. C. Greubel, Sep 27 2018

Mathematica

Table[ Simplify[ ((2 + Sqrt@2)^n - (2 - Sqrt@2)^n)^2/8], {n, 0, 19}] (* Robert G. Wilson v, Sep 24 2008 *)
CoefficientList[Series[x (1 + 2 x) / ((1 - 2 x) (1 - 12 x + 4 x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 06 2018 *)
LinearRecurrence[{14, -28, 8}, {0, 1, 16}, 20] (* Harvey P. Dale, Apr 12 2020 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r2>:=NumberField(x^2-2); [ Integers()!a: a in [ ((2+r2)^n-(2-r2)^n)^2/8: n in [0..19] ] ]; // Klaus Brockhaus, Oct 20 2008
(Magma) I:=[0, 1, 16]; [n le 3 select I[n] else 14*Self(n-1)-28*Self(n-2)+8*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Feb 05 2018
(PARI) x='x+O('x^30); concat([0], Vec(x*(1+2*x)/((1-2*x)*(1-12*x+4*x^2)) )) \\ G. C. Greubel, Sep 27 2018

Crossrefs

Sequence in context: A016173 A005747 A103721 * A093060 A153885 A016226

Adjacent sequences: A144841 A144842 A144843 * A144845 A144846 A144847

Keyword

nonn,easy

Author

Al Hakanson (hawkuu(AT)gmail.com), Sep 22 2008

Status

approved