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A088320 a(n) = 10*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 5.

%I A088320 #27 Dec 14 2022 06:12:55
%S A088320 1,5,51,515,5201,52525,530451,5357035,54100801,546365045,5517751251,
%T A088320 55723877555,562756526801,5683289145565,57395647982451,
%U A088320 579639768970075,5853793337683201,59117573145802085,597029524795704051,6029412821102842595,60891157735824130001,614940990179344142605
%N A088320 a(n) = 10*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 5.
%H A088320 G. C. Greubel, <a href="/A088320/b088320.txt">Table of n, a(n) for n = 0..990</a>
%H A088320 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A088320 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,1).
%H A088320 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A088320 a(n) = 10*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 5.
%F A088320 a(n) = ((5+sqrt(26))^n + (5-sqrt(26))^n)/2.
%F A088320 a(n) = A086927(n)/2.
%F A088320 Lim_{n -> oo} a(n+1)/a(n) = (5+sqrt(26)) = 10.099019... .
%F A088320 Lim_{n -> oo} a(n)/a(n+1) = 1/(5+sqrt(26)) = (sqrt(26)-5) = 0.099019... .
%F A088320 From _Paul Barry_, Nov 15 2003: (Start)
%F A088320 E.g.f.: exp(5*x)*cosh(sqrt(26)*x).
%F A088320 a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*26^k*5^(n-2*k).
%F A088320 a(n) = (-i)^n * T(n, 5*i), with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)
%F A088320 G.f.: (1-5*x)/(1-10*x-x^2). - _R. J. Mathar_, Sep 11 2008
%t A088320 LinearRecurrence[{10,1},{1,5},31] (* _Harvey P. Dale_, Dec 25 2021 *)
%o A088320 (Magma) [n le 2 select 5^(n-1) else 10*Self(n-1) + Self(n-2): n in [1..31]]; // _G. C. Greubel_, Dec 12 2022
%o A088320 (SageMath)
%o A088320 A088320=BinaryRecurrenceSequence(10,1,1,5)
%o A088320 [A088320(n) for n in range(31)] # _G. C. Greubel_, Dec 12 2022
%Y A088320 Cf. A041040, A041043, A064019, A077392, A086927.
%K A088320 easy,nonn
%O A088320 0,2
%A A088320 Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Nov 06 2003