%I #20 Mar 06 2023 12:42:29

%S 5,9,23,55,133,321,775,1871,4517,10905,26327,63559,153445,370449,

%T 894343,2159135,5212613,12584361,30381335,73347031,177075397,

%U 427497825,1032071047,2491639919,6015350885,14522341689,35060034263,84642410215

%N a(n) = ((5 + 2*sqrt(2))*(1 + sqrt(2))^n + (5 - 2*sqrt(2))*(1 - sqrt(2))^n)/2.

%C Binomial transform of A163888. Inverse binomial transform of A163608.

%H G. C. Greubel, <a href="/A163607/b163607.txt">Table of n, a(n) for n = 0..1000</a>

%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) = 5, a(1) = 9.

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

%F a(n) = 5*A000129(n+1) - A000129(n). - _R. J. Mathar_, Nov 08 2013

%F E.g.f.: exp(x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - _G. C. Greubel_, Jul 29 2017

%F a(n) = 2*A001333(n) + A001333(n+2). - _Philippe Deléham_, Mar 06 2023

%t LinearRecurrence[{2, 1}, {5, 9}, 50] (* _G. C. Greubel_, Jul 29 2017 *)

%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+2*r)*(1+r)^n+(5-2*r)*(1-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 06 2009

%o (PARI) x='x+O('x^50); Vec((5-x)/(1-2*x-x^2)) \\ _G. C. Greubel_, Jul 29 2017

%Y Cf. A163608, A163888.

%Y Cf. A000129, A001333.

%K nonn,easy

%O 0,1

%A Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 06 2009