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%I #13 Jun 29 2023 23:37:47
%S 1,4,22,136,868,5584,35992,232096,1496848,9653824,62262112,401558656,
%T 2589848128,16703198464,107727106432,694784866816,4481007870208,
%U 28900214293504,186391635645952,1202130942232576,7753131181401088
%N a(n) = ((4 + sqrt 6)^n + (4 - sqrt 6)^n)/2.
%C Binomial transform of A084120. Sequences defined by a(n) = ((A + sqrt(B))^n + (A - sqrt(B))^n)/2 have recurrences a(n) = 2*A*a(n-1) + (B - A^2)*a(n-2) and generating functions g.f.: (1-Ax)/(1-2Ax+(A^2-B)x^2). - _R. J. Mathar_, Nov 01 2008
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8, -10).
%F From _R. J. Mathar_, _Klaus Brockhaus_ and _Philippe Deléham_, Nov 01 2008: (Start)
%F a(n) = 8*a(n-1) - 10*a(n-2).
%F G.f.: (1-4x)/(1-8x+10x^2). (End)
%F a(n) = (Sum_{k=0..n} A098158(n,k)*4^(2*k)*6^(n-k))/4^n. - _Philippe Deléham_, Nov 06 2008
%e a(3) = 136.
%e a(4) = ((4 + sqrt(6))^4 + (4 - sqrt(6))^4)/2 = 4^4 + 6*sqrt(6)^2*4^2 + sqrt(6)^4 = 4^4 + 6*6*4^2 + 6^2 = 868. - _Klaus Brockhaus_, Nov 01 2008
%t Table[MatrixPower[{{4,2},{3,4}},n][[1]][[1]],{n,0,44}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2010 *)
%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r6>:=NumberField(x^2-6); S:=[ ((4+r6)^n+(4-r6)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Nov 01 2008
%K nonn,easy
%O 0,2
%A Al Hakanson (hawkuu(AT)gmail.com), Oct 27 2008
%E More terms from _Klaus Brockhaus_ and _R. J. Mathar_, Nov 01 2008
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