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%I #33 Sep 08 2022 08:45:11
%S 2,9,83,756,6887,62739,571538,5206581,47430767,432083484,3936182123,
%T 35857722591,326655685442,2975758891569,27108485709563,
%U 246952130277636,2249677658208287,20494051054152219,186696137145578258
%N a(n) = 9*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 9.
%C a(n+1)/a(n) converges to (9 + sqrt(85))/2.
%C For more information about this type of recurrence follow the Khovanova link and see A054413 and A086902. - _Johannes W. Meijer_, Jun 12 2010
%H Vincenzo Librandi, <a href="/A087798/b087798.txt">Table of n, a(n) for n = 0..1000</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/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,1).
%F a(n) = ((9 + sqrt(85))/2)^n + ((9 - sqrt(85))/2)^n.
%F G.f.: (2 - 9*x)/(1 - 9*x - x^2). - _Philippe Deléham_, Nov 02 2008
%F From _Johannes W. Meijer_, Jun 12 2010: (Start)
%F a(2n+1) = 9*A097840(n), a(2n) = A099373(n).
%F a(3n+1) = A041150(5n), a(3n+2) = A041150(5n+3), a(3n+3) = 2*A041150(5n+4).
%F Lim_{k->infinity} a(n+k)/a(k) = (A087798(n) + A099371(n)*sqrt(85))/2.
%F Lim_{n->infinity} A087798(n)/A099371(n) = sqrt(85). (End)
%e a(4) = 9*a(3) + a(2) = 9*756 + 83 = 6887.
%t RecurrenceTable[{a[0] == 2, a[1] == 9, a[n] == 9 a[n-1] + a[n-2]}, a, {n, 30}] (* _Vincenzo Librandi_, Sep 19 2016 *)
%t LinearRecurrence[{9,1}, {2,9}, 30] (* _G. C. Greubel_, Nov 07 2018 *)
%o (Magma) I:=[2,9]; [n le 2 select I[n] else 9*Self(n-1)+Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 19 2016
%o (PARI) x='x+O('x^30); Vec((2-9*x)/(1-9*x-x^2)) \\ _G. C. Greubel_, Nov 07 2018
%Y Cf. A014511.
%K easy,nonn
%O 0,1
%A Nikolay V. Kosinov, Dmitry V. Poljakov (kosinov(AT)unitron.com.ua), Oct 10 2003
%E More terms from _Ray Chandler_, Nov 06 2003
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