A164594 a(n) = ((5 + sqrt(18))*(4 + sqrt(8))^n + (5 - sqrt(18))*(4 - sqrt(8))^n)/2.

5, 32, 216, 1472, 10048, 68608, 468480, 3198976, 21843968, 149159936, 1018527744, 6954942464, 47491317760, 324291002368, 2214397476864, 15120851795968, 103251634552832, 705046262054912, 4814357020016640, 32874486063693824, 224481032349417472

Offset: 0

Author

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

Keywords

• nonn

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (8,-8).

Crossrefs

Cf. A101386, A164737, A164595.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=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

Maple

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

Mathematica

CoefficientList[Series[(5-8*x)/(1-8*x+8*x^2), {x,0,25}], x] (* G. C. Greubel, Aug 12 2017 *)
Table[(2*Sqrt[2])^n*(3*ChebyshevU[n, Sqrt[2]] + 2*ChebyshevT[n, Sqrt[2]]), {n, 0, 25}] (* G. C. Greubel, Apr 21 2020 *)
LinearRecurrence[{8,-8},{5,32},30] (* Harvey P. Dale, Jul 09 2022 *)

PARI

my(x='x+O('x^25)); Vec((5-8*x)/(1-8*x+8*x^2)) \\ G. C. Greubel, Aug 12 2017

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

Formula

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

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

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

a(n) = (2*sqrt(2))^n * (3*ChebyshevU(n, sqrt(2)) + 2*ChebyshevT(n, sqrt(2))). - G. C. Greubel, Apr 21 2020

Extensions

Extended by Klaus Brockhaus and R. J. Mathar Aug 24 2009