%I #12 Sep 08 2022 08:45:46

%S 1,10,66,388,2180,12008,65544,356240,1932304,10471072,56716320,

%T 307135552,1663055936,9004549760,48753614976,263965223168,

%U 1429171175680,7737856281088,41894453789184,226825642378240,1228082785977344

%N a(n) = ((1 + 3*sqrt(2))*(4 + sqrt(2))^n + (1 - 3*sqrt(2))*(4 - sqrt(2))^n)/2.

%C Binomial transform of A163614. Fourth binomial transform of A163864. Inverse binomial transform of A163616.

%H Harvey P. Dale, <a href="/A163615/b163615.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8, -14).

%F a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 10.

%F G.f.: (1+2*x)/(1-8*x+14*x^2).

%F E.g.f.: exp(4*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - _G. C. Greubel_, Jul 30 2017

%t LinearRecurrence[{8,-14},{1,10},30] (* _Harvey P. Dale_, Jun 11 2014 *)

%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+3*r)*(4+r)^n+(1-3*r)*(4-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 06 2009

%o (PARI) x='x+O('x^50); Vec((1+2*x)/(1-8*x+14*x^2)) \\ _G. C. Greubel_, Jul 30 2017

%Y Cf. A163614, A163864, A163616.

%K nonn

%O 0,2

%A Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 06 2009