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%I #22 Sep 08 2022 08:45:47
%S 1,10,44,216,1040,5024,24256,117120,565504,2730496,13184000,63657984,
%T 307367936,1484103680,7165886464,34599960576,167063388160,
%U 806653394944,3894867132416,18806082109440,90803796967424,438439516307456
%N a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 10.
%C Binomial transform of A083100. Second binomial transform of A164683. Inverse binomial transform of A054490.
%H G. C. Greubel, <a href="/A164607/b164607.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..164 from Vincenzo Librandi)
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4, 4).
%F a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 10.
%F a(n) = ((2+4*sqrt(2))*(2+2*sqrt(2))^n + (2-4*sqrt(2))*(2-2*sqrt(2))^n)/4.
%F G.f.: (1+6*x)/(1-4*x-4*x^2).
%F G.f.: G(0)/(2*x) - 1/x, where G(k)= 1 + 1/(1 - x*(8*k-1)/(x*(8*k+7) - (1-x)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013
%F E.g.f.: exp(2*x)*(cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x)). - _G. C. Greubel_, Aug 10 2017
%t LinearRecurrence[{4,4},{1,10},40] (* _Harvey P. Dale_, Jun 28 2011 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((2+4*r)*(2+2*r)^n+(2-4*r)*(2-2*r)^n)/4: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 22 2009
%o (PARI) x='x+O('x^50); Vec((1+6*x)/(1-4*x-4*x^2)) \\ _G. C. Greubel_, Aug 10 2017
%Y Cf. A083100, A164683, A054490.
%K nonn,easy
%O 0,2
%A Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009
%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 22 2009
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