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A290471
Number of minimal edge covers in the n-wheel graph.
1
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
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
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