login
a(n) = J(n) if n odd, or 4*J(n) if n even, where J = Jacobsthal numbers A001045.
1

%I #13 Jan 12 2020 11:26:17

%S 0,1,4,3,20,11,84,43,340,171,1364,683,5460,2731,21844,10923,87380,

%T 43691,349524,174763,1398100,699051,5592404,2796203,22369620,11184811,

%U 89478484,44739243,357913940,178956971,1431655764,715827883,5726623060,2863311531,22906492244

%N a(n) = J(n) if n odd, or 4*J(n) if n even, where J = Jacobsthal numbers A001045.

%H Colin Barker, <a href="/A270797/b270797.txt">Table of n, a(n) for n = 0..1000</a>

%H H. Bruhn, L. Gellert, J. Günther, <a href="https://arxiv.org/abs/1503.03390">Jacobsthal numbers in generalised Petersen graphs</a>, arXiv preprint arXiv:1503.03390 [math.CO], 2015.

%H H. Bruhn, L. Gellert, J. Günther, <a href="http://eurocomb2015.b.uib.no/files/2015/08/endm1943.pdf">Jacobsthal numbers in generalised Petersen graphs</a>, Electronic Notes in Discrete Math., 2015.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-4).

%F From _Colin Barker_, Apr 01 2016: (Start)

%F a(n) = (-3+3*(-2)^n-5*(-1)^n+5*2^n)/6.

%F a(n) = (2^(n+3)-8)/6 for n even.

%F a(n) = (2^(n+1)+2)/6 for n odd.

%F a(n) = 5*a(n-2)-4*a(n-4) for n>3.

%F G.f.: x*(1+4*x-2*x^2) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)).

%F (End)

%o (PARI) concat(0, Vec(x*(1+4*x-2*x^2)/((1-x)*(1+x)*(1-2*x)*(1+2*x)) + O(x^50))) \\ _Colin Barker_, Apr 01 2016

%Y Cf. A001045.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Mar 31 2016