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A127222 a(n) = 3^n*pentanacci(n) or (3^n)*A023424(n-1).

%I A127222 #20 Sep 19 2024 21:57:11
%S A127222 3,27,189,1215,7533,41553,247131,1463103,8640837,50959287,300264165,
%T A127222 1771292853,10447598619,61618989627,363414767589,2143339285311,
%U A127222 12641143135581,74555586323649,439717218548643,2593383067853775,15295369041550269,90209719910309895
%N A127222 a(n) = 3^n*pentanacci(n) or (3^n)*A023424(n-1).
%H A127222 G. C. Greubel, <a href="/A127222/b127222.txt">Table of n, a(n) for n = 1..1000</a>
%H A127222 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,9,27,81,243).
%F A127222 a(n) = Trace of matrix [({3,3,3,3,3},{3,0,0,0,0},{0,3,0,0,0},{0,0,3,0,0},{0,0,0,3,0})^n].
%F A127222 a(n) = 3^n * Trace of matrix [({1,1,1,1,1},{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0})^n].
%F A127222 G.f.: -3*x*(1 +6*x +27*x^2 +108*x^3 +405*x^4)/(-1 +3*x +9*x^2 +27*x^3 +81*x^4 +243*x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009
%F A127222 a(n) = 3*a(n-1)+9*a(n-2)+27*a(n-3)+81*a(n-4)+243*a(n-5). - _Colin Barker_, Sep 02 2013
%t A127222 Table[Tr[MatrixPower[3*{{1, 1, 1, 1, 1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, x]], {x, 1, 20}]
%t A127222 LinearRecurrence[{3, 9, 27, 81, 243}, {3, 27, 189, 1215, 7533}, 50] (* _G. C. Greubel_, Dec 19 2017 *)
%o A127222 (PARI) x='x+O('x^30); Vec(-3*x*(1 +6*x +27*x^2 +108*x^3 +405*x^4)/(-1 +3*x +9*x^2 +27*x^3 +81*x^4 +243*x^5)) \\ _G. C. Greubel_, Dec 19 2017
%o A127222 (Magma) I:=[3, 27, 189, 1215, 7533]; [n le 5 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3) + 81*Self(n-4) + 243*Self(n-5): n in [1..30]]; // _G. C. Greubel_, Dec 19 2017
%Y A127222 Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127220, A127221.
%Y A127222 Cf. A023424, A074048.
%K A127222 nonn,easy
%O A127222 1,1
%A A127222 _Artur Jasinski_, Jan 09 2007
%E A127222 G.f. proposed by Maksym Voznyy checked and corrected by _R. J. Mathar_, Sep 16 2009
%E A127222 Definition corrected by _R. J. Mathar_, Sep 17 2009
%E A127222 More terms from _Colin Barker_, Sep 02 2013