A290471 Number of minimal edge covers in the n-wheel graph.

1, 5, 7, 17, 31, 52, 99, 169, 301, 536, 947, 1698, 3043, 5482, 9919, 17993, 32743, 59713, 109099, 199634, 365719, 670605, 1230547, 2259286, 4149841, 7624904, 14013499, 25759764, 47358741, 87077529, 160120891, 294453673, 541509775, 995888884, 1831584239

Offset

2,2

Comments

The n-wheel graph is well defined for n>=4. Sequence extended to n=2 using recurrence. This gives values for a single edge and for a wheel with two spokes connected by two rim edges. - Andrew Howroyd, Aug 04 2017

Links

Andrew Howroyd, Table of n, a(n) for n = 2..200

Eric Weisstein's World of Mathematics, Minimal Edge Cover

Eric Weisstein's World of Mathematics, Wheel Graph

Index entries for linear recurrences with constant coefficients, signature (1, 3, 1, -5, -5, 0, 4, 3, 1).

Formula

From Andrew Howroyd, Aug 04 2017: (Start)

a(n) = a(n-1)+3*a(n-2)+a(n-3)-5*a(n-4)-5*a(n-5)+4*a(n-7)+3*a(n-8)+a(n-9) for n>10.

G.f.: x^2*(1 + x)*(1 + 3*x - 4*x^2 - 2*x^3 - 5*x^4 - 2*x^5 - x^6)/((1 - x^2 - x^3)^2*(1 - x - x^2 - x^3)).

(End)

Mathematica

LinearRecurrence[{1, 5, 7, 17, 31, 52, 99, 169, 301}, {1, 3, 1, -5, -5, 0, 4, 3, 1}, 20]
CoefficientList[Series[(-1 - 4 x + x^2 + 6 x^3 + 7 x^4 + 7 x^5 + 3 x^6 + x^7)/((-1 + x^2 + x^3)^2 (-1 + x + x^2 + x^3)), {x, 0, 20}], x]
Table[RootSum[-1 - # - #^2 + #^3 &, #^n &] - RootSum[-1 - # + #^3 &, -#^(1 + n) + #^(3 + n) &] + n RootSum[-1 - # + #^3 &, -#^(n + 1) - 8 #^(n + 2) + 13 #^(n + 3) &]/23, {n, 20}]

Prog

(PARI) Vec((1 + x)*(1 + 3*x - 4*x^2 - 2*x^3 - 5*x^4 - 2*x^5 - x^6)/((1 - x^2 - x^3)^2*(1 - x - x^2 - x^3))+O(x^40)) \\ Andrew Howroyd, Aug 04 2017

Crossrefs

Sequence in context: A126142 A216560 A019340 * A261792 A166977 A272717

Adjacent sequences: A290468 A290469 A290470 * A290472 A290473 A290474

Keyword

nonn

Author

Eric W. Weisstein, Aug 03 2017

Extensions

a(2)-a(3) and terms a(14) and beyond from Andrew Howroyd, Aug 04 2017

Status

approved