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A284709 Number of maximal matchings in the wheel graph on n nodes. 2
2, 1, 4, 3, 10, 10, 20, 28, 42, 63, 92, 132, 194, 273, 394, 555, 786, 1105, 1550, 2166, 3022, 4200, 5832, 8073, 11162, 15400, 21218, 29187, 40098, 55013, 75392, 103199, 141122, 192786, 263128, 358820, 488918, 665667, 905656, 1231308, 1672962, 2271605 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Linear recurrence used to extrapolate to a(1), a(2), a(3).
LINKS
Tomislav Doslic, I. Zubac, Counting maximal matchings in linear polymers, Ars Mathematica Contemporanea 11 (2016) 255-276. See Prop. 7.2.
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Maximal Independent Edge Set
Eric Weisstein's World of Mathematics, Wheel Graph
FORMULA
a(n) = 3*a(n-2) + 2*a(n-3) - 3*a(n-4) - 4*a(n-5) + 2*a(n-7) + a(n-8).
G.f.: (x*(-2 - x + 2*x^2 + 4*x^3 - 2*x^4 - 4*x^5 + x^7))/((-1 + x^2)*(-1 + x^2 + x^3)^2).
a(n) = (n-1)*Padovan(n+3)+1-(-1)^n, where Padovan(k) = A000931(k). (Eee Doslic et al.) - N. J. A. Sloane, Apr 24 2017
MAPLE
A000931 := proc(n) option remember; if n = 0 then 1 elif n <= 2 then 0 else procname(n-2)+procname(n-3); fi; end;
psi:=n->A000931(n+6);
f:=n->n*psi(n-2)+1+(-1)^n;
[seq(f(n), n=0..40)]; # Produces the sequence with an offset of 0 - N. J. A. Sloane, Apr 24 2017
MATHEMATICA
LinearRecurrence[{0, 3, 2, -3, -4, 0, 2, 1}, {2, 1, 4, 3, 10, 10, 20, 28, 42, 63, 92}, 37] (* Eric W. Weisstein, Apr 01 2017 *)
Padovan[n_] := RootSum[-1 - # + #^3 &, 5 #^n - 6 #^(n + 1) + 4 #^(n + 2) &]/23; Table[(n - 1) Padovan[n + 3] - (-1)^n + 1, {n, 20}] (* Eric W. Weisstein, Dec 30 2017 *)
CoefficientList[Series[(-2 - x + 2 x^2 + 4 x^3 - 2 x^4 - 4 x^5 + x^7)/((-1 + x^2) (-1 + x^2 + x^3)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 30 2017 *)
PROG
(PARI) Vec(x*(2 + x - 2*x^2 - 4*x^3 + 2*x^4 + 4*x^5 - x^7) / ((1 - x)*(1 + x)*(1 - x^2 - x^3)^2) + O(x^50)) \\ Colin Barker, Apr 25 2017
CROSSREFS
Cf. A000931.
Sequence in context: A109195 A258310 A217927 * A307365 A032662 A185413
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Apr 01 2017
STATUS
approved

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Last modified March 19 06:16 EDT 2024. Contains 370952 sequences. (Running on oeis4.)