A163611 - OEIS

%I #11 Sep 08 2022 08:45:46

%S 5,29,175,1083,6805,43141,274895,1756707,11244485,72040589,461782735,

%T 2960893803,18987935125,121778793781,781065429935,5009742042387,

%U 32132915535365,206105088378749,1321993826474095,8479521232029723

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

%C Binomial transform of A163610. Fifth binomial transform of A163888.

%H G. C. Greubel, <a href="/A163611/b163611.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,-23).

%F a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 5, a(1) = 29.

%F G.f.: (5-21*x)/(1-10*x+23*x^2).

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

%t LinearRecurrence[{10, -23}, {5, 29}, 50] (* _G. C. Greubel_, Jul 29 2017 *)

%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+2*r)*(5+r)^n+(5-2*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 06 2009

%o (PARI) x='x+O('x^50); Vec((5-21*x)/(1-10*x+23*x^2)) \\ _G. C. Greubel_, Jul 29 2017

%Y Cf. A163610, A163888.

%K nonn

%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