%I #46 Feb 16 2025 08:33:58
%S 0,0,0,0,0,6,24,41,60,84,112,144,180,220,264,312,364,420,480,544,612,
%T 684,760,840,924,1012,1104,1200,1300,1404,1512,1624,1740,1860,1984,
%U 2112,2244,2380,2520,2664,2812,2964,3120,3280,3444,3612,3784,3960,4140,4324,4512
%N a(n) is the maximum number of 5-cycles possible in an n-vertex planar graph.
%C All the terms are even numbers except for a(7) = 41 which is also the only prime.
%C For n >= 5, also the number of 5-cycles in the (n-2)-dipyramidal graph. - _Eric W. Weisstein_, Dec 07 2023
%H Stefano Spezia, <a href="/A328180/b328180.txt">Table of n, a(n) for n = 0..10000</a>
%H Debarun Ghosh, Ervin Győri, Oliver Janzer, Addisu Paulos, Nika Salia, and Oscar Zamora, <a href="https://arxiv.org/abs/2004.01162">The maximum number of induced C_5's in a planar graph</a>, arXiv:2004.01162 [math.CO], 2020.
%H Ervin Győri, Addisu Paulos, Nika Salia, Casey Tompkins, and Oscar Zamora, <a href="https://arxiv.org/abs/1909.13532">The Maximum Number of Pentagons in a Planar Graph</a>, arXiv:1909.13532 [math.CO], 2019.
%H S. L. Hakimi and E. F. Schmeichel, <a href="https://doi.org/10.1002/jgt.3190030108">On the number of cycles of length k in a maximal planar graph</a>, J. Graph Theory 3 (1979): 69-86.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DipyramidalGraph.html">Dipyramidal Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F O.g.f.: x^5*(-6 - 6*x + 13*x^2 - 3*x^3 - 3*x^4 + x^5)/(-1 + x)^3.
%F E.g.f.: x^7/5040 - x^5/20 - x^4/6 + 2*exp(x)*x^2 - 8*exp(x)*x - 4*x + 12*exp(x) - 12.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 10.
%F a(n) = 0 for n < 5, a(5) = 6, a(6) = 24, a(7) = 41, a(n) = 2*n^2 - 10*n + 12 for n > 7 (see Theorem 1 in Győri et al.).
%F a(n) = A046092(n-3) for n > 7.
%F a(n) = A106232(n-2) for n > 7.
%p gf := (1/5040)*x^7-(1/20)*x^5-(1/6)*x^4+2*exp(x)*x^2-8*exp(x)*x-4*x+12*exp(x)-12; ser := series(gf, x, 51); seq(factorial(n)*coeff(ser, x, n), n = 0..50)
%t Join[{0,0,0,0,0,6,24,41},Table[2n^2-10n+12,{n,8,50}]]
%t LinearRecurrence[{3,-3,1},{0,0,0,0,0,6,24,41,60,84,112},60] (* _Harvey P. Dale_, Jan 10 2022 *)
%o (Magma) I:=[0, 0, 0, 0, 0, 6, 24, 41, 60, 84, 112]; [n le 11 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..51]];
%o (PARI) concat([0, 0, 0, 0, 0], Vec(x^5*(-6-6*x+13*x^2-3*x^3-3*x^4+x^5)/(-1+x)^3+O(x^51)))
%o (Magma) R<x>:=PowerSeriesRing(Integers(),51); [0,0,0,0,0] cat Coefficients(R!(x^5*(-6-6*x+13*x^2-3*x^3-3*x^4+x^5)/(-1+x)^3)); // _Marius A. Burtea_, Oct 16 2019
%Y Cf. A005843, A046092, A106232, A185721.
%K nonn,easy,changed
%O 0,6
%A _Stefano Spezia_, Oct 06 2019