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Triameter of the n-cube-connected cycle graph.
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%I #16 Apr 19 2023 02:37:57

%S 13,20,25,32,36,44,48,56,60,68,72,80,84,92,96,104,108,116,120,128,132,

%T 140,144,152,156,164,168,176,180,188,192,200,204,212,216,224,228,236,

%U 240,248,252,260,264,272,276,284,288,296,300,308,312,320,324,332,336

%N Triameter of the n-cube-connected cycle graph.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cube-ConnectedCycleGraph.html">Cube-Connected Cycle Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphTriameter.html">Graph Triameter</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, -1).

%F a(2n) = 12*n - 4, a(2n+1) = 12*n for n > 2. - _Andrew Howroyd_, Apr 15 2018

%F G.f.: (x^5-x^4-8*x^2+7*x+13)/(x^3-x^2-x+1). - _Georg Fischer_, Nov 17 2022

%e From _Andrew Howroyd_, Apr 16 2018: (Start)

%e Vertices can be represented by a pair (k,w) where k in 0..n-1 is the current index and w is an n bit word.

%e In the following, words are shown with index zero as the rightmost bit. Example vertices are given with maximal total distance between them. Similar constructions can be used for all n. These constructions are not unique.

%e Case n=5: with vertices v1=(0,00000), v2=(0,01100), v3=(0,11011)

%e d(v1,v2)=2+5=7, d(v1,v3)=4+5=9, d(v2,v3)=4+5=9

%e total distance is 7 + 9 + 9 = 25 = a(5).

%e Case n=7: with vertices v1=(0,0000000), v2=(0,0011000), v3=(3,1111111)

%e d(v1,v2)=2+7=9, d(v1,v3)=7+8=15, d(v2,v3)=5+7=12

%e total distance is 9 + 15 + 12 = 36 = a(7).

%e Case n=10: with vertices v1=(0,0000000000), v2=(0,000010000), v3=(5,1111111111)

%e d(v1,v2)=1+10=11, d(v1,v3)=10+13=23, d(v2,v3)=9+13=22

%e total distance is 11 + 23 + 22 = 56 = a(10).

%e (End)

%t CoefficientList[Series[(x^5-x^4-8*x^2+7*x+13)/(x^3-x^2-x+1),{x,0, 40}],x] (* _Georg Fischer_, Nov 17 2022 *)

%K nonn,easy

%O 3,1

%A _Eric W. Weisstein_, Apr 06 2018

%E Terms a(8) and beyond from _Andrew Howroyd_, Apr 15 2018