A164608 Expansion of (1+4*x)/(1-8*x+8*x^2).

1, 12, 88, 608, 4160, 28416, 194048, 1325056, 9048064, 61784064, 421888000, 2880831488, 19671547904, 134325731328, 917233467392, 6263261888512, 42768227368960, 292039723843584, 1994171971796992, 13617057983627264

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Binomial transform of A054490. Fourth binomial transform of A164683. Inverse binomial transform of A164609.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..149 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (8,-8).

Crossrefs

Cf. A054490, A164683, A164609.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+4*r)*(4+2*r)^n+(2-4*r)*(4-2*r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 22 2009

Mathematica

LinearRecurrence[{8,-8}, {1,12,88}, 50] (* G. C. Greubel, Aug 10 2017 *)

PARI

x='x+O('x^50); Vec((1+4*x)/(1-8*x+8*x^2)) \\ G. C. Greubel, Aug 10 2017

Formula

a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 1, a(1) = 12.

a(n) = A057084(n) + 4*A057084(n-1).

a(n) = ((2+4*sqrt(2))*(4+2*sqrt(2))^n + (2-4*sqrt(2))*(4-2*sqrt(2))^n)/4.

E.g.f.: exp(4*x)*(cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 10 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 22 2009