A298648 Number of smallest coverings of the n-dipyramidal graph by maximal cliques.

1, 4, 30, 12, 98, 28, 270, 60, 682, 124, 1638, 252, 3810, 508, 8670, 1020, 19418, 2044, 42966, 4092, 94162, 8188, 204750, 16380, 442314, 32764, 950214, 65532, 2031554, 131068, 4325310, 262140, 9174970, 524284, 19398582, 1048572, 40894386, 2097148, 85983150

Offset: 3

Author

Eric W. Weisstein, Jun 18 2018

Keywords

• nonn
• easy

Links

• Muniru A Asiru, Table of n, a(n) for n = 3..700
• Eric Weisstein's World of Mathematics, Dipyramidal Graph
• Eric Weisstein's World of Mathematics, Maximal Clique
• Eric Weisstein's World of Mathematics, Minimum Clique Covering
• Index entries for linear recurrences with constant coefficients, signature (0,6,0,-13,0,12,0,-4)

Crossrefs

Cf. A110654 (clique covering number of the n-dipyramidal graph).

Programs

Maple

seq(coeff(series((1+4*x+24*x^2-12*x^3-69*x^4+8*x^5+60*x^6-20*x^8)/(1-3*x^2+2*x^4)^2, x,n+1),x,n),n=0..38); # Muniru A Asiru, Jul 02 2018

Mathematica

Join[{1}, Table[If[Mod[n, 2] == 0, 2, n] (2^Ceiling[n/2] - 2), {n, 4, 20}]]
Join[{1}, Table[2 (1 + (-1)^n) (2^(n/2 - 1) - 1) + (1 - (-1)^n) (2^((n - 1)/2) - 1) n, {n, 4, 20}]]
Join[{1}, LinearRecurrence[{0, 6, 0, -13, 0, 12, 0, -4}, {4, 30, 12, 98, 28, 270, 60, 682}, 20]]
CoefficientList[Series[(1 + 4 x + 24 x^2 - 12 x^3 - 69 x^4 + 8 x^5 + 60 x^6 - 20 x^8)/(1 - 3 x^2 + 2 x^4)^2, {x, 0, 20}], x]

PARI

a(n)={if(n==3, 1, if(n%2, n, 2)*(2^ceil(n/2)-2))} \\ Andrew Howroyd, Jun 27 2018

PARI

Vec(x^3*(1 + 4*x + 24*x^2 - 12*x^3 - 69*x^4 + 8*x^5 + 60*x^6 - 20*x^8) / ((1 - x)^2*(1 + x)^2*(1 - 2*x^2)^2) + O(x^45)) \\ Colin Barker, Jul 20 2019

Formula

a(2*k) = 2^(k+1) - 4, a(2*k-1) = (2*k-1)*(2^k - 2) for k > 2. - Andrew Howroyd, Jun 27 2018

From Colin Barker, Jul 20 2019: (Start)

G.f.: x^3*(1 + 4*x + 24*x^2 - 12*x^3 - 69*x^4 + 8*x^5 + 60*x^6 - 20*x^8) / ((1 - x)^2*(1 + x)^2*(1 - 2*x^2)^2).

a(n) = (1 + (-1)^n)*(-2+2^(n/2)) + ((-1+(-1)^n)*(sqrt(2) - 2^(n/2))*n)/sqrt(2) for n>3.

a(n) = 6*a(n-2) - 13*a(n-4) + 12*a(n-6) - 4*a(n-8) for n>8.

(End)

Extensions

Terms a(19) and beyond from Andrew Howroyd, Jun 27 2018