%I #9 Sep 08 2022 08:45:46
%S 3,17,103,637,3963,24697,153983,960197,5987763,37339937,232854103,
%T 1452093517,9055353003,56469795337,352149479663,2196028088597,
%U 13694580432483,85400334485297,532562291125063,3321094649662237
%N a(n) = ((3+sqrt(5))*(4+sqrt(5))^n + (3-sqrt(5))*(4-sqrt(5))^n)/2.
%C Binomial transform of A098648 without initial 1. Fourth binomial transform of A163114. Inverse binomial transform of A163065.
%H G. C. Greubel, <a href="/A163064/b163064.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 (8,-11).
%F a(n) = 8*a(n-1) - 11*a(n-2) for n > 1; a(0) = 3, a(1) = 17.
%F G.f.: (3-7*x)/(1-8*x+11*x^2).
%t CoefficientList[Series[(3-7*x)/(1-8*x+11*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{8,-11}, {3,17}, 30] (* _G. C. Greubel_, Dec 22 2017 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((3+r)*(4+r)^n+(3-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 21 2009
%o (Magma) I:=[3,17]; [n le 2 select I[n] else 8*Self(n-1) - 11*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 22 2017
%o (PARI) x='x+O('x^30); Vec((3-7*x)/(1-8*x+11*x^2)) \\ _G. C. Greubel_, Dec 22 2017
%Y Cf. A098648, A163114, A163065.
%K nonn
%O 0,1
%A Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009
%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Jul 21 2009