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a(n) = A005248(n) - n.
0

%I #30 Oct 14 2024 15:59:22

%S 2,2,5,15,43,118,316,836,2199,5769,15117,39592,103670,271430,710633,

%T 1860483,4870831,12752026,33385264,87403784,228826107,599074557,

%U 1568397585,4106118220,10749957098,28143753098,73681302221,192900153591

%N a(n) = A005248(n) - n.

%C Also distinct compositions of the wheel graph W_n. - _Ralf Stephan_, Jan 02 2003

%H A. Knopfmacher and M. E. Mays, <a href="http://www.emis.de/journals/INTEGERS/papers/b4/b4.Abstract.html">Graph Compositions. I: Basic Enumeration</a>, Integers 1(2001), #A04.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,5,-1).

%F a(n) = 3*a(n-1) - a(n-2) + n - 1.

%F G.f.: (2 - 8*x + 11*x^2 - 4*x^3)/((1-3*x+x^2)*(1-x)^2).

%F a(n) = Lucas(2*n) - n. - _G. C. Greubel_, Oct 12 2019

%F E.g.f.: 2*exp(3*x/2)*cosh(sqrt(5)*x/2) - exp(x)*x. - _Stefano Spezia_, Oct 14 2024

%p with(combinat); seq(fibonacci(2*n+1)+fibonacci(2*n-1)-n, n=0..50); # _G. C. Greubel_, Oct 12 2019

%t Table[LucasL[2*n]-n, {n,0,50}] (* _G. C. Greubel_, Oct 12 2019 *)

%o (PARI) a(n)=fibonacci(2*n+1)+fibonacci(2*n-1)-n

%o (Magma) [Lucas(2*n) - n: n in [0..50]]; // _G. C. Greubel_, Oct 12 2019

%o (Sage) [lucas_number2(2*n, 1,-1) - n for n in range(50)] # _G. C. Greubel_, Oct 12 2019

%o (GAP) List([0..50], n-> Lucas(1,-1, 2*n)[2] - n ); # _G. C. Greubel_, Oct 12 2019

%Y Cf. A000032, A000045, A005248.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_