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%I #29 Feb 08 2022 22:54:45
%S 1,6,26,116,516,2296,10216,45456,202256,899936,4004256,17816896,
%T 79276096,352738176,1569504896,6983495936,31072993536,138258966016,
%U 615181851136,2737245336576,12179345048576,54191870867456
%N a(n) = 4*a(n-1) + 2*a(n-2) for n > 1; a(0) = 1, a(1) = 6.
%C Binomial transform of A123011. Inverse binomial transform of A164550.
%C INVERT transform of the sequence (1, 5, 5*3, 5*3^2, 5*3^3, 5*3^4, ...); i.e., of (1, 5, 15, 45, 135, 405, ...). The sequence can also be obtained by extracting the upper left terms in matrix powers of [(1,5); (1,3)]. - _Gary W. Adamson_, Jul 31 2016
%C The sequence is A090017 (1, 4, 18, 80, 356, ...) convolved with (1, 2, 0, 0, 0, ...). Also, the upper left terms extracted from matrix powers of [(1,5); (1,3)]. - _Gary W. Adamson_, Aug 20 2016
%H Harvey P. Dale, <a href="/A164549/b164549.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 (4,2).
%F a(n) = ((3+2*sqrt(6))*(2+sqrt(6))^n + (3-2*sqrt(6))*(2-sqrt(6))^n)/6.
%F G.f.: (1+2*x)/(1-4*x-2*x^2).
%F a(n) = (i*sqrt(2))^n*(ChebyshevU(n, -i*sqrt(2)) - sqrt(2)*i*ChebyshevU(n-1, -i*sqrt(2))). - _G. C. Greubel_, Jul 16 2021
%t LinearRecurrence[{4,2},{1,6},30] (* _Harvey P. Dale_, Mar 16 2013 *)
%t CoefficientList[Series[(1 +2x)/(1 -4x -2x^2), {x, 0, 24}], x] (* _Michael De Vlieger_, Aug 02 2016 *)
%o (Magma) [ n le 2 select 5*n-4 else 4*Self(n-1)+2*Self(n-2): n in [1..22] ];
%o (PARI) Vec((1+2*x)/(1-4*x-2*x^2) + O(x^30)) \\ _Michel Marcus_, Feb 04 2016
%o (Sage) [(i*sqrt(2))^n*(chebyshev_U(n, -i*sqrt(2)) - sqrt(2)*i*chebyshev_U(n-1, -i*sqrt(2))) for n in (0..30)] # _G. C. Greubel_, Jul 16 2021
%Y Cf. A123011, A164550.
%Y Cf. A084057, A108306.
%K nonn
%O 0,2
%A _Klaus Brockhaus_, Aug 15 2009
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