A164594 - OEIS

%I #21 Sep 08 2022 08:45:47

%S 5,32,216,1472,10048,68608,468480,3198976,21843968,149159936,

%T 1018527744,6954942464,47491317760,324291002368,2214397476864,

%U 15120851795968,103251634552832,705046262054912,4814357020016640,32874486063693824,224481032349417472

%N a(n) = ((5 + sqrt(18))*(4 + sqrt(8))^n + (5 - sqrt(18))*(4 - sqrt(8))^n)/2.

%C Binomial transform of A101386. Fourth binomial transform of A164737. Inverse binomial transform of A164595.

%H G. C. Greubel, <a href="/A164594/b164594.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..100 from Vincenzo Librandi)

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

%F a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 5, a(1) = 32.

%F G.f.: (5-8*x)/(1-8*x+8*x^2).

%F E.g.f.: exp(4*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - _G. C. Greubel_, Aug 12 2017

%F a(n) = (2*sqrt(2))^n * (3*ChebyshevU(n, sqrt(2)) + 2*ChebyshevT(n, sqrt(2))). - _G. C. Greubel_, Apr 21 2020

%p A164594:= (n) -> simplify( (2*sqrt(2))^n*(5*ChebyshevU(n, sqrt(2)) - 2*sqrt(2)*ChebyshevU(n-1, sqrt(2))) ); seq( A164594(n), n = 0..25); # _G. C. Greubel_, Apr 21 2020

%t CoefficientList[Series[(5-8*x)/(1-8*x+8*x^2), {x,0,25}], x] (* _G. C. Greubel_, Aug 12 2017 *)

%t Table[(2*Sqrt[2])^n*(3*ChebyshevU[n, Sqrt[2]] + 2*ChebyshevT[n, Sqrt[2]]), {n, 0, 25}] (* _G. C. Greubel_, Apr 21 2020 *)

%t LinearRecurrence[{8,-8},{5,32},30] (* _Harvey P. Dale_, Jul 09 2022 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+3*r)*(4+2*r)^n+(5-3*r)*(4-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 24 2009

%o (PARI) my(x='x+O('x^25)); Vec((5-8*x)/(1-8*x+8*x^2)) \\ _G. C. Greubel_, Aug 12 2017

%o (Sage) [(2*sqrt(2))^n*(5*chebyshev_U(n, sqrt(2)) - 2*sqrt(2)*chebyshev_U(n-1, sqrt(2))) for n in (0..25)] # _G. C. Greubel_, Apr 21 2020

%Y Cf. A101386, A164737, A164595.

%K nonn

%O 0,1

%A Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

%E Extended by _Klaus Brockhaus_ and _R. J. Mathar_ Aug 24 2009