A164605 a(n) = ((1+4*sqrt(2))*(4+2*sqrt(2))^n + (1-4*sqrt(2))*(4-2*sqrt(2))^n)/2.

1, 20, 152, 1056, 7232, 49408, 337408, 2304000, 15732736, 107429888, 733577216, 5009178624, 34204811264, 233565061120, 1594881998848, 10890535501824, 74365228023808, 507797540175872, 3467458497216512, 23677287656325120

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Binomial transform of A164604. Fourth binomial transform of A164702. Inverse binomial transform of A164606.

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. A164604, A164702, A164606.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(4+2*r)^n+(1-4*r)*(4-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 23 2009

Mathematica

LinearRecurrence[{8,-8},{1,20},30] (* Harvey P. Dale, Mar 24 2015 *)

PARI

Vec((1+12*x)/(1-8*x+8*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2011

Formula

a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 1, a(1) = 20.

G.f.: (1+12*x)/(1-8*x+8*x^2).

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

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 23 2009