%I #35 Aug 24 2024 22:41:46

%S 2,13,171,2236,29239,382343,4999698,65378417,854919119,11179326964,

%T 146186169651,1911599532427,24996980091202,326872340718053,

%U 4274337409425891,55893258663254636,730886700031736159,9557420359075824703,124977351368017457298,1634262988143302769577

%N a(n) = 13*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 13.

%C For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413. - _Johannes W. Meijer_, Jun 12 2010

%H G. C. Greubel, <a href="/A088316/b088316.txt">Table of n, a(n) for n = 0..890</a>

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

%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>

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

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,1).

%F a(n) = ((13+sqrt(173))/2)^n + ((13-sqrt(173))/2)^n.

%F Lim_{n -> oo} a(n+1)/a(n) = (13 + sqrt(173))/2.

%F Lim_{n -> oo} a(n)/a(n+1) = 2/(13+sqrt(173)).

%F G.f.: (2-13*x)/(1-13*x-x^2). - _Philippe Deléham_, Nov 02 2008

%F From _Johannes W. Meijer_, Jun 12 2010: (Start)

%F a(2*n+1) = 13*A097845(n).

%F a(3*n+1) = A041318(5n), a(3n+2) = A041318(5n+3), a(3n+3) = 2*A041318(5n+4).

%F Limit_{k->oo} a(n+k)/a(k) = (A088316(n) + A140455(n)*sqrt(173))/2.

%F Limit_{n->oo} A088316(n)/A140455(n) = sqrt(173). (End)

%t LinearRecurrence[{13,1}, {2,13}, 31] (* _Stefano Spezia_, Sep 20 2022 *)

%o (Magma) I:=[2,13]; [n le 2 select I[n] else 13*Self(n-1) +Self(n-2): n in [1..31]]; // _G. C. Greubel_, Dec 13 2022

%o (SageMath)

%o A088316=BinaryRecurrenceSequence(13,1,2,13)

%o [A088316(n) for n in range(31)] # _G. C. Greubel_, Dec 13 2022

%Y Cf. A006905, A041318, A054413, A086902, A088316, A097845, A140455.

%K nonn,easy,changed

%O 0,1

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