A164588 - OEIS

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

%S 1,9,73,577,4529,35481,277817,2174993,17027041,133295529,1043495593,

%T 8168931937,63949894289,500627099961,3919122796697,30680567267633,

%U 240180585132481,1880236207775049,14719292130498313,115228905772807297,902061091509601649

%N a(n) = ((3 + sqrt(18))*(5 + sqrt(8))^n + (3 - sqrt(18))*(5 - sqrt(8))^n)/6.

%C Binomial transform of A057084. Second binomial transform of A002315. Third binomial transform of A108051 without initial 0. Fourth binomial transform of A096980. Fifth binomial transform of A094015.

%H G. C. Greubel, <a href="/A164588/b164588.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 (10,-17).

%F a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 9.

%F G.f.: (1-x)/(1-10*x+17*x^2).

%F E.g.f.: (1/3)*exp(5*x)*(3*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - _G. C. Greubel_, Aug 12 2017

%t LinearRecurrence[{10,-17},{1,9},30] (* _Harvey P. Dale_, Sep 11 2016 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((3+3*r)*(5+2*r)^n+(3-3*r)*(5-2*r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 24 2009

%o (PARI) x='x+O('x^50); Vec((1-x)/(1-10*x+17*x^2)) \\ _G. C. Greubel_, Aug 12 2017

%Y Cf. A057084, A002315, A108051, A096980, A094015.

%K nonn

%O 0,2

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

%E Extended by _Klaus Brockhaus_ and _R. J. Mathar_ Aug 24 2009