A164592 a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 8.

1, 8, 63, 494, 3869, 30292, 237147, 1856506, 14533561, 113775008, 890679543, 6972620294, 54584650709, 427311962092, 3345180558867, 26187502233106, 205006952830321, 1604881990340408, 12563701705288623, 98354023217099294, 769957303181086349, 6027554637120175492

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Binomial transform of A164591. Fifth binomial transform of A096886.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..144 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (10,-17).

Crossrefs

Cf. A164591, A096886.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+3*r)*(5+2*r)^n+(4-3*r)*(5-2*r)^n)/8: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

Mathematica

CoefficientList[Series[(1 - 2*z)/(17*z^2 - 10*z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 12 2011 *)
LinearRecurrence[{10,-17},{1,8},30] (* Harvey P. Dale, Oct 14 2012 *)

PARI

my(x='x+O('x^30)); Vec((1-2*x)/(1-10*x+17*x^2)) \\ G. C. Greubel, Aug 12 2017

Formula

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 8.

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

G.f.: (1-2*x)/(1-10*x+17*x^2).

E.g.f.: (1/4)*exp(5*x)*(4*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017

Extensions

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