OFFSET
0,6
COMMENTS
All the terms are even numbers except for a(7) = 41 which is also the only prime.
For n >= 5, also the number of 5-cycles in the (n-2)-dipyramidal graph. - Eric W. Weisstein, Dec 07 2023
LINKS
Stefano Spezia, Table of n, a(n) for n = 0..10000
Debarun Ghosh, Ervin Győri, Oliver Janzer, Addisu Paulos, Nika Salia, and Oscar Zamora, The maximum number of induced C_5's in a planar graph, arXiv:2004.01162 [math.CO], 2020.
Ervin Győri, Addisu Paulos, Nika Salia, Casey Tompkins, and Oscar Zamora, The Maximum Number of Pentagons in a Planar Graph, arXiv:1909.13532 [math.CO], 2019.
S. L. Hakimi and E. F. Schmeichel, On the number of cycles of length k in a maximal planar graph, J. Graph Theory 3 (1979): 69-86.
Eric Weisstein's World of Mathematics, Dipyramidal Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
O.g.f.: x^5*(-6 - 6*x + 13*x^2 - 3*x^3 - 3*x^4 + x^5)/(-1 + x)^3.
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.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 10.
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.).
a(n) = A046092(n-3) for n > 7.
a(n) = A106232(n-2) for n > 7.
MAPLE
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)
MATHEMATICA
Join[{0, 0, 0, 0, 0, 6, 24, 41}, Table[2n^2-10n+12, {n, 8, 50}]]
LinearRecurrence[{3, -3, 1}, {0, 0, 0, 0, 0, 6, 24, 41, 60, 84, 112}, 60] (* Harvey P. Dale, Jan 10 2022 *)
PROG
(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]];
(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)))
(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
KEYWORD
nonn,easy
AUTHOR
Stefano Spezia, Oct 06 2019
STATUS
approved