A088320 - OEIS

%I #27 Dec 14 2022 06:12:55

%S 1,5,51,515,5201,52525,530451,5357035,54100801,546365045,5517751251,

%T 55723877555,562756526801,5683289145565,57395647982451,

%U 579639768970075,5853793337683201,59117573145802085,597029524795704051,6029412821102842595,60891157735824130001,614940990179344142605

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

%H G. C. Greubel, <a href="/A088320/b088320.txt">Table of n, a(n) for n = 0..990</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,1).

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.

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

%F a(n) = ((5+sqrt(26))^n + (5-sqrt(26))^n)/2.

%F a(n) = A086927(n)/2.

%F Lim_{n -> oo} a(n+1)/a(n) = (5+sqrt(26)) = 10.099019... .

%F Lim_{n -> oo} a(n)/a(n+1) = 1/(5+sqrt(26)) = (sqrt(26)-5) = 0.099019... .

%F From _Paul Barry_, Nov 15 2003: (Start)

%F E.g.f.: exp(5*x)*cosh(sqrt(26)*x).

%F a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*26^k*5^(n-2*k).

%F 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 G.f.: (1-5*x)/(1-10*x-x^2). - _R. J. Mathar_, Sep 11 2008

%t LinearRecurrence[{10,1},{1,5},31] (* _Harvey P. Dale_, Dec 25 2021 *)

%o (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 (SageMath)

%o A088320=BinaryRecurrenceSequence(10,1,1,5)

%o [A088320(n) for n in range(31)] # _G. C. Greubel_, Dec 12 2022

%Y Cf. A041040, A041043, A064019, A077392, A086927.

%K easy,nonn

%O 0,2

%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Nov 06 2003