A020877 -id:A020877 - cp's OEIS frontend

A290470 Number of minimal edge covers in the n-Moebius ladder.

3, 7, 15, 59, 143, 367, 1039, 2755, 7395, 20007, 53727, 144635, 389535, 1048159, 2821535, 7595267, 20443523, 55029319, 148125295, 398712379, 1073232175, 2888862159, 7776059055, 20931132355, 56341155043, 151655701607, 408217663167, 1098815603707, 2957725352255

Offset: 1

Eric W. Weisstein, Aug 03 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Edge Cover
Eric Weisstein's World of Mathematics, Minimal Edge Cover
Eric Weisstein's World of Mathematics, Moebius Ladder
Index entries for linear recurrences with constant coefficients, signature (1, 2, 6, 2, 2, -2, -2, -1, 1).

Cf. A020866, A020877, A020878, A284710.

Table[2 Cos[n Pi/2] - RootSum[-1 + # + #^2 + #^3 &, #^n &] +
  RootSum[1 - 2 #^2 - 2 #^3 + #^4 &, #^n &], {n, 20}]
LinearRecurrence[{1, 2, 6, 2, 2, -2, -2, -1, 1}, {3, 7, 15, 59, 143, 367, 1039, 2755, 7395}, 20]
CoefficientList[Series[-(((1 + x) (-3 - x - x^2 + x^3) (-1 - 4 x^3 + 3 x^4))/((1 + x^2) (-1 - x - x^2 + x^3) (1 - 2 x - 2 x^2 + x^4))), {x, 0, 20}], x]

Vec((1 + x)*(1 + 4*x^3 - 3*x^4)*(3 + x + x^2 - x^3)/((1 + x^2)*(1 + x + x^2 - x^3)*(1 - 2*x - 2*x^2 + x^4)) + O(x^30)) \\ Andrew Howroyd, Aug 04 2017

From Andrew Howroyd, Aug 04 2017: (Start)
a(n) = a(n-1) + 2*a(n-2) + 6*a(n-3) + 2*a(n-4) + 2*a(n-5) - 2*a(n-6) - 2*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: x*(1 + x)*(1 + 4*x^3 - 3*x^4)*(3 + x + x^2 - x^3)/((1 + x^2)*(1 + x + x^2 - x^3)*(1 - 2*x - 2*x^2 + x^4)).
(End)

a(1)-a(2) and terms a(9) and beyond from Andrew Howroyd, Aug 04 2017

A302232 Triangle T(n,k) of the numbers of k-matchings in the n-Moebius ladder (0 <= k <= n, n > 2).

Original entry on oeis.org

1, 9, 18, 6, 1, 12, 42, 44, 7, 1, 15, 75, 145, 95, 13, 1, 18, 117, 336, 420, 192, 18, 1, 21, 168, 644, 1225, 1085, 371, 31, 1, 24, 228, 1096, 2834, 3880, 2588, 696, 47, 1, 27, 297, 1719, 5652, 10656, 11097, 5823, 1278, 78, 1, 30, 375, 2540, 10165, 24626, 35645, 29380, 12535, 2310, 123

Offset: 3

Eric W. Weisstein, Apr 03 2018

Initial terms in each row match those in A061702.

			As polynomials sum(k=0..n) x^k*T(n, k):
1 + 9*x + 18*x^2 + 6*x^3,
1 + 12*x + 42*x^2 + 44*x^3 + 7*x^4,
1 + 15*x + 75*x^2 + 145*x^3 + 95*x^4 + 13*x^5,
1 + 18*x + 117*x^2 + 336*x^3 + 420*x^4 + 192*x^5 + 18*x^6,
...

Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Eric Weisstein's World of Mathematics, Moebius Ladder

Row sums are A020877.

Cf. A061702.

CoefficientList[LinearRecurrence[{1 + x, 2 x (1 + x), -(-1 + x) x^2, -x^4}, {1 + 3 x, 1 + 6 x + 3 x^2, 1 + 9 x + 18 x^2 + 6 x^3, 1 + 12 x + 42 x^2 + 44 x^3 + 7 x^4}, {3, 10}], x] // Flatten
CoefficientList[CoefficientList[Series[-((-1 - 9 x - 18 x^2 - 6 x^3 - 2 x z - 15 x^2 z - 20 x^3 z - x^4 z - x^2 z^2 - 5 x^3 z^2 + 4 x^4 z^2 + 6 x^5 z^2 + x^4 z^3 + 6 x^5 z^3 + 3 x^6 z^3)/((1 + x z) (1 - z - 2 x z - x z^2 + x^3 z^3))), {z, 0, 10}], z], x] // Flatten

G.f.: -((z^2*(-1 - 9*x - 18*x^2 - 6*x^3 - 2*x*z - 15*x^2*z - 20*x^3*z - x^4*z - x^2*z^2 - 5*x^3*z^2 + 4*x^4*z^2 + 6*x^5*z^2 + x^4*z^3 + 6*x^5*z^3 + 3*x^6*z^3))/((1 + x*z)*(1 - z - 2*x*z - x*z^2 + x^3*z^3))).

Writing t(n, x) = sum(k=0..n) x^k*T(n, k), t(n, x) == (1 + x)*t(n-1, x) + 2*x*(1 + x)*t(n-2, x) -(-1 + x)*x^2*t(n-3, x) -x^4*t(n-4, x).

cp's OEIS Frontend

A290470 Number of minimal edge covers in the n-Moebius ladder.

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