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%I #63 Sep 08 2022 08:44:39
%S 0,1,-2,7,-20,61,-182,547,-1640,4921,-14762,44287,-132860,398581,
%T -1195742,3587227,-10761680,32285041,-96855122,290565367,-871696100,
%U 2615088301,-7845264902,23535794707,-70607384120,211822152361,-635466457082,1906399371247
%N a(n) = (1 - (-3)^n)/4.
%C q-integers for q=-3.
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^n*charpoly(A,0). - _Milan Janjic_, Jan 27 2010
%C Pisano period lengths: 1, 2, 1, 4, 4, 2, 3, 8, 1, 4, 10, 4, 6, 6, 4, 16, 16, 2, 9, 4, ... - _R. J. Mathar_, Aug 10 2012
%H T. D. Noe, <a href="/A014983/b014983.txt">Table of n, a(n) for n = 0..200</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=927">Encyclopedia of Combinatorial Structures 927</a>
%H László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Nemeth/nemeth6.html">The trinomial transform triangle</a>, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also <a href="https://arxiv.org/abs/1807.07109">arXiv:1807.07109</a> [math.NT], 2018.
%H R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SULANKE/sulanke.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,3)
%F a(n) = a(n-1) + (-3)^(n-1).
%F G.f.: x/((1-x)*(1+3*x)).
%F a(n) = -(-1)^n*A015518(n).
%F a(n) = the (1, 2)-th element of M^n, where M = ((1, 1, 1, -2), (1, 1, -2, 1), (1, -2, 1, 1), (-2, 1, 1, 1)). - _Simone Severini_, Nov 25 2004
%F a(0)=0, a(1)=1, a(n) = -2*a(n-1) + 3*a(n-2) for n>1. - _Philippe Deléham_, Sep 19 2009
%F From _Sergei N. Gladkovskii_, Apr 29 2012: (Start)
%F G.f. A(x) = G(0)/4; G(k) = 1 - 1/(3^(2*k) - 3*x*3^(4*k)/(3*x*3^(2*k) + 1/(1 + 1/(3*3^(2*k) - 3^(3)*x*3^(4*k)/(3^2*x*3^(2*k) - 1/G(k+1)))))); (continued fraction, 3rd kind, 6-step).
%F E.g.f. E(x) = G(0)/4; G(k) = 1 - 1/(9^k - 3*x*81^k/(3*x*9^k + (2*k+1)/(1 + 1/(3*9^k - 27*x*81^k/(9*x*9^k - (2*k+2)/G(k+1)))))); (continued fraction, 3rd kind, 6-step). (End)
%F a(n) = A084222(n) - 1. - _Filip Zaludek_, Nov 19 2016
%F E.g.f.: sinh(x)*cosh(x)*exp(-x). - _Ilya Gutkovskiy_, Nov 20 2016
%p a:=n->sum ((-3)^j, j=0..n): seq(a(n), n=-1..25); # _Zerinvary Lajos_, Dec 16 2008
%t nn = 25; CoefficientList[Series[x/((1 - x)*(1 + 3*x)), {x, 0, nn}], x] (* _T. D. Noe_, Jun 21 2012 *)
%t Table[(1 - (-3)^n)/4, {n, 0, 27}] (* _Michael De Vlieger_, Nov 23 2016 *)
%o (PARI) a(n)=(1-(-3)^n)/4
%o (Sage) [gaussian_binomial(n,1,-3) for n in range(0,27)] # _Zerinvary Lajos_, May 28 2009
%o (Magma) [(1-(-3)^n)/4: n in [0..30]]; // _G. C. Greubel_, May 26 2018
%Y Cf. A077925, A014985, A014986, A014987, A014989, A014990, A014991, A014992, A014993, A014994. - _Zerinvary Lajos_, Dec 16 2008
%K sign,easy
%O 0,3
%A _Olivier Gérard_
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