A290700 Number of minimal edge covers in the n-prism graph.

1, 5, 25, 49, 141, 389, 1009, 2761, 7441, 19925, 53769, 144721, 389325, 1048325, 2821665, 7594761, 20444065, 55029413, 148124153, 398713969, 1073231821, 2888859781, 7776063377, 20931130057, 56341150641, 151655712629, 408217654249, 1098815597201

Offset

1,2

Comments

The n-prism graph is well defined for n >= 3. Sequence extended to n = 1 using recurrence. - Andrew Howroyd, Aug 10 2017

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Minimal Edge Cover

Eric Weisstein's World of Mathematics, Prism Graph

Index entries for linear recurrences with constant coefficients, signature (1, 2, 6, 2, 2, -2, -2, -1, 1).

Formula

From Andrew Howroyd, Aug 10 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 + 4*x + 18*x^2 + 8*x^3 + 10*x^4 - 12*x^5 - 14*x^6 - 8*x^7 + 9*x^8)/((1 - 2*x - 2*x^2 + x^4)*(1 + x + x^2 - x^3)*(1 + x^2)).

(End)

Mathematica

Table[2 Cos[n Pi/2] + RootSum[-1 + # + #^2 + #^3 &, #^n &] -
  RootSum[1 - 2 #^2 - 2 #^3 + #^4 &, -2 #^(n + 2) - 2 #^(n + 3) + #^(n + 4) &], {n, 20}]
LinearRecurrence[{1, 2, 6, 2, 2, -2, -2, -1, 1}, {1, 5, 25, 49, 141, 389, 1009, 2761, 7441}, 20]
CoefficientList[Series[-( (1 + 4 x + 18 x^2 + 8 x^3 + 10 x^4 - 12 x^5 - 14 x^6 - 8 x^7 + 9 x^8)/((1 + x^2) (-1 - x - x^2 + x^3) (1 - 2 x - 2 x^2 + x^4))), {x, 0, 20}], x]

Prog

(PARI)
Vec((1 + 4*x + 18*x^2 + 8*x^3 + 10*x^4 - 12*x^5 - 14*x^6 - 8*x^7 + 9*x^8)/((1 - 2*x - 2*x^2 + x^4)*(1 + x + x^2 - x^3)*(1 + x^2))+O(x^30)) \\ Andrew Howroyd, Aug 10 2017

Crossrefs

Cf. A123304.

Sequence in context: A298041 A074493 A262760 * A136914 A136913 A136911

Adjacent sequences: A290697 A290698 A290699 * A290701 A290702 A290703

Keyword

nonn

Author

Eric W. Weisstein, Aug 09 2017

Extensions

a(1)-a(2) and terms a(9) and beyond from Andrew Howroyd, Aug 10 2017

Status

approved