A164303 - OEIS

%I #12 Sep 08 2022 08:45:47

%S 3,11,25,61,147,355,857,2069,4995,12059,29113,70285,169683,409651,

%T 988985,2387621,5764227,13916075,33596377,81108829,195814035,

%U 472736899,1141287833,2755312565,6651912963,16059138491,38770189945,93599518381

%N a(n) = 2*a(n-1) + a(n-2) for n > 1; a(0) = 3, a(1) = 11.

%C Binomial transform of A164654. Inverse binomial transform of A164304.

%H G. C. Greubel, <a href="/A164303/b164303.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..218 from Vincenzo Librandi)

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).

%F a(n) = 2*a(n-1)+a(n-2) for n > 1; a(0) = 3, a(1) = 11.

%F a(n) = ((3+4*sqrt(2))*(1+sqrt(2))^n + (3-4*sqrt(2))*(1-sqrt(2))^n)/2.

%F G.f.: (3+5*x)/(1-2*x-x^2).

%F E.g.f.: (3*cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(x). - _G. C. Greubel_, Sep 13 2017

%t LinearRecurrence[{2,1}, {3,11}, 50] (* or *) CoefficientList[Series[(3 + 5*x)/(1 - 2*x - x^2), {x,0,50}], x] (* _G. C. Greubel_, Sep 13 2017 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((3+4*r)*(1+r)^n+(3-4*r)*(1-r)^n)/2: n in [0..28] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 20 2009

%o (PARI) x='x+O('x^50); Vec((3+5*x)/(1-2*x-x^2)) \\ _G. C. Greubel_, Sep 13 2017

%Y Cf. A164654, A164304.

%K nonn,easy

%O 0,1

%A Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 20 2009