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%I #13 Sep 08 2022 08:45:46
%S 4,7,22,58,160,436,1192,3256,8896,24304,66400,181408,495616,1354048,
%T 3699328,10106752,27612160,75437824,206099968,563075584,1538351104,
%U 4202853376,11482408960,31370524672,85705867264,234152783872,639717302272,1747740172288,4774914949120
%N a(n) = ((4+sqrt(3))*(1+sqrt(3))^n + (4-sqrt(3))*(1-sqrt(3))^n)/2.
%C Binomial transform of A162766. Inverse binomial transform of A077236.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,2).
%F a(n) = 2*a(n-1) + 2*a(n-2) for n > 1; a(0) = 4, a(1) = 7.
%F G.f.: (4-x)/(1-2*x-2*x^2).
%p seq((1/2)*simplify((4+sqrt(3))*(1+sqrt(3))^n+(4-sqrt(3))*(1-sqrt(3))^n), n = 0 .. 27); # _Emeric Deutsch_, Jul 16 2009
%t LinearRecurrence[{2,2},{4,7},30] (* _Harvey P. Dale_, Sep 21 2018 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((4+r)*(1+r)^n+(4-r)*(1-r)^n)/2: n in [0..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 13 2009
%Y Cf. A162766, A077236.
%K nonn
%O 0,1
%A Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009
%E Edited by _Klaus Brockhaus_, _Paolo P. Lava_ and Emeric Deutsch, Jul 13 2009
%E Two different extensions were received. This version was rechecked by _N. J. A. Sloane_, Jul 19 2009