A164608 - OEIS

%I #20 Mar 19 2024 09:32:23

%S 1,12,88,608,4160,28416,194048,1325056,9048064,61784064,421888000,

%T 2880831488,19671547904,134325731328,917233467392,6263261888512,

%U 42768227368960,292039723843584,1994171971796992,13617057983627264

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

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

%H G. C. Greubel, <a href="/A164608/b164608.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..149 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) = 1, a(1) = 12.

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

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

%F 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

%t LinearRecurrence[{8,-8}, {1,12,88}, 50] (* _G. C. Greubel_, Aug 10 2017 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=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

%o (PARI) x='x+O('x^50); Vec((1+4*x)/(1-8*x+8*x^2)) \\ _G. C. Greubel_, Aug 10 2017

%Y Cf. A054490, A164683, A164609.

%K nonn,easy

%O 0,2

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

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 22 2009