A298822 Number of minimum edge covers in the n-dipyramidal graph.

1, 2, 21, 8, 85, 18, 217, 32, 441, 50, 781, 72, 1261, 98, 1905, 128, 2737, 162, 3781, 200, 5061, 242, 6601, 288, 8425, 338, 10557, 392, 13021, 450, 15841, 512, 19041, 578, 22645, 648, 26677, 722, 31161, 800, 36121, 882, 41581, 968, 47565, 1058, 54097, 1152

Offset

1,2

Comments

The size of a minimum edge cover is given by floor((n + 3)/2). - Andrew Howroyd, Jun 26 2018

Links

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

Eric Weisstein's World of Mathematics, Dipyramidal Graph

Eric Weisstein's World of Mathematics, Minimum Edge Cover

Index entries for linear recurrences with constant coefficients, signature (0, 4, 0, -6, 0, 4, 0, -1).

Formula

From Andrew Howroyd, Jun 26 2018: (Start)

a(2*n) = 2*n^2, a(2*n-1) = (2*n-1)*(2*n^2 - 1).

a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8) for n > 8.

G.f.: x*(1 + 2*x + 17*x^2 + 7*x^4 - 2*x^5 - x^6)/((1 - x)^4*(1 + x)^4). (End)

a(n) = n*(n^2 + 3*n - 1 - (-1)^n*(n^2 + n - 1))/4. - Eric W. Weisstein, Jun 27 2018

Mathematica

Table[n (n^2 + 3 n - 1 - (-1)^n (n^2 + n - 1))/4, {n, 20}]
LinearRecurrence[{0, 4, 0, -6, 0, 4, 0, -1}, {1, 2, 21, 8, 85, 18, 217, 32}, 20]
CoefficientList[Series[(1 + 2 x + 17 x^2 + 7 x^4 - 2 x^5 - x^6)/(-1 + x^2)^4, {x, 0, 20}], x]

Prog

(PARI) a(n)={n*if(n%2, 2*(n\2+1)^2-1, n\2)} \\ Andrew Howroyd, Jun 26 2018

Crossrefs

Cf. A296995, A297713.

Sequence in context: A358520 A105666 A058261 * A072397 A077208 A321534

Adjacent sequences: A298819 A298820 A298821 * A298823 A298824 A298825

Keyword

nonn

Author

Eric W. Weisstein, Jun 18 2018

Extensions

a(1)-a(2) and terms a(10) and beyond from Andrew Howroyd, Jun 26 2018

Status

approved