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

%I A298648 #35 Feb 16 2025 08:33:53
%S A298648 1,4,30,12,98,28,270,60,682,124,1638,252,3810,508,8670,1020,19418,
%T A298648 2044,42966,4092,94162,8188,204750,16380,442314,32764,950214,65532,
%U A298648 2031554,131068,4325310,262140,9174970,524284,19398582,1048572,40894386,2097148,85983150
%N A298648 Number of smallest coverings of the n-dipyramidal graph by maximal cliques.
%H A298648 Muniru A Asiru, <a href="/A298648/b298648.txt">Table of n, a(n) for n = 3..700</a>
%H A298648 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DipyramidalGraph.html">Dipyramidal Graph</a>
%H A298648 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>
%H A298648 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumCliqueCovering.html">Minimum Clique Covering</a>
%H A298648 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-13,0,12,0,-4)
%F A298648 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
%F A298648 From _Colin Barker_, Jul 20 2019: (Start)
%F A298648 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).
%F A298648 a(n) = (1 + (-1)^n)*(-2+2^(n/2)) + ((-1+(-1)^n)*(sqrt(2) - 2^(n/2))*n)/sqrt(2) for n>3.
%F A298648 a(n) = 6*a(n-2) - 13*a(n-4) + 12*a(n-6) - 4*a(n-8) for n>8.
%F A298648 (End)
%p A298648 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
%t A298648 Join[{1}, Table[If[Mod[n, 2] == 0, 2, n] (2^Ceiling[n/2] - 2), {n, 4, 20}]]
%t A298648 Join[{1}, Table[2 (1 + (-1)^n) (2^(n/2 - 1) - 1) + (1 - (-1)^n) (2^((n - 1)/2) - 1) n, {n, 4, 20}]]
%t A298648 Join[{1}, LinearRecurrence[{0, 6, 0, -13, 0, 12, 0, -4}, {4, 30, 12, 98, 28, 270, 60, 682}, 20]]
%t A298648 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]
%o A298648 (PARI) a(n)={if(n==3, 1, if(n%2, n, 2)*(2^ceil(n/2)-2))} \\ _Andrew Howroyd_, Jun 27 2018
%o A298648 (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
%Y A298648 Cf. A110654 (clique covering number of the n-dipyramidal graph).
%K A298648 nonn,easy
%O A298648 3,2
%A A298648 _Eric W. Weisstein_, Jun 18 2018
%E A298648 Terms a(19) and beyond from _Andrew Howroyd_, Jun 27 2018