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%I #53 Sep 08 2022 08:46:13
%S 3,2,2,5,10,17,29,51,90,158,277,486,853,1497,2627,4610,8090,14197,
%T 24914,43721,76725,134643,236282,414646,727653,1276942,2240877,
%U 3932465,6900995,12110402,21252274,37295141,65448410,114853953,201554637,353703731,620706778
%N a(n) = a(n-1) + a(n-2) + a(n-4).
%C Also the number of maximal independent vertex sets (and minimal vertex covers) in the n-gear graph. - _Eric W. Weisstein_, May 25 2017
%C Also the number of chordless cycles in the n-antiprism graph for n >= 4. - _Eric W. Weisstein_, Jan 02 2018
%D R. K. Guy, Letter to N. J. A. Sloane, Feb 05 1986.
%H Reinhard Zumkeller, <a href="/A259967/b259967.txt">Table of n, a(n) for n = 0..1000</a>
%H Nazim Fatès, Biswanath Sethi, Sukanta Das, <a href="https://hal.inria.fr/hal-01571847">On the reversibility of ECAs with fully asynchronous updating: the recurrence point of view</a>, Preprint, hal-01571847, 2017.
%H R. K. Guy, <a href="/A005251/a005251.pdf">Letter to N. J. A. Sloane, Feb 1986</a>
%H Bojan Vučković and Miodrag Živković, <a href="https://www.researchgate.net/publication/312219294">Row Space Cardinalities Above 2^(n - 2) + 2^(n - 3)</a>, ResearchGate, January 2017, p. 3.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AntiprismGraph.html">Antiprism Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalIndependentVertexSet.html">Maximal Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MinimalVertexCover.html">Minimal Vertex Cover</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1).
%F G.f.: (x-1)*(x-3) / (1 -2*x +x^2 -x^3). - _R. J. Mathar_, Jul 15 2015
%F a(n) = -4*A005314(n) +3*A005314(n+1) +A005314(n-1). - _R. J. Mathar_, Jul 15 2015
%F a(n) = Sum_{i=1..3} r_i^n where r_i are the roots of x^3-2*x^2+x-1. - _Robert Israel_, Jul 18 2016
%F a(n) = A109377(n-2) for n > 1. - _Georg Fischer_, Oct 09 2018
%p f:= gfun:-rectoproc({-a(n+3)+2*a(n+2)-a(n+1)+a(n), a(0) = 3, a(1) = 2, a(2) = 2},a(n),remember):
%p map(f, [$0..50]); # _Robert Israel_, Jul 18 2016
%t Abs @ CoefficientList[Series[(x - 1) (x - 3)/(-1 + 2 x - x^2 + x^3), {x, 0, 36}], x] (* _Michael De Vlieger_, Jul 18 2016 *)
%t LinearRecurrence[{2, -1, 1}, {2, 2, 5}, 20] (* _Eric W. Weisstein_, May 25 2017 *)
%t Table[RootSum[-1 + # - 2 #^2 + #^3 &, #^n &], {n, 20}] (* _Eric W. Weisstein_, May 25 2017 *)
%t RootSum[-1 + # - 2 #^2 + #^3 &, #^Range[0, 20] &] (* _Eric W. Weisstein_, Jan 02 2018 *)
%o (Haskell)
%o a259967 n = a259967_list !! n
%o a259967_list = 3 : 2 : 2 : 5 : zipWith3 (((+) .) . (+))
%o a259967_list (drop 2 a259967_list) (drop 3 a259967_list)
%o -- _Reinhard Zumkeller_, Jul 12 2015
%o (PARI) x='x+O('x^50); Vec((x-1)*(x-3)/(1-2*x+x^2-x^3)) \\ _G. C. Greubel_, May 24 2017
%o (Magma) I:=[3,2,2,5]; [n le 4 select I[n] else Self(n-1)+Self(n-2)+Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Sep 26 2017
%Y Cf. A109377, A259968, A259969.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_, Jul 11 2015
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