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The hyper-Wiener index of the cyclic phenylene with n hexagons (n>=3).
1

%I #11 Mar 29 2019 04:06:30

%S 1062,2760,5715,10386,17304,27072,40365,57930,80586,109224,144807,

%T 188370,241020,303936,378369,465642,567150,684360,818811,972114,

%U 1145952,1342080,1562325,1808586,2082834,2387112,2723535,3094290

%N The hyper-Wiener index of the cyclic phenylene with n hexagons (n>=3).

%C a(3), a(4), ..., a(16) have been checked by the direct computation of the hyper-Wiener index (using Maple).

%D Y. Alizadeh, S. Klavzar, The Wiener dimension of a graph (unpublished manuscript).

%H G. Cash, S. Klavzar, M. Petkovsek, <a href="https://doi.org/10.1021/ci0100999">Three methods for calculation of the hyper-Wiener index of a molecular graph</a>, J. Chem. Inf. Comput. Sci. 42, 2002, 571-576.

%F a(n) = (3/2)n(2n^3 +15n^2 + 45n -88).

%F G.f.: 3z^3(354-850z+845z^2-403z^3+78z^4)/(1-z)^5.

%F The Hosoya polynomial of the cyclic phenylene with n hexagons is [n*t^n*(t^5+3t^4+5t^3+5t^2+3t+1) - n(t^8+t^7+t^6+t^5+2t^3+4t^2+8t)]/(t-1).

%p a := proc (n) options operator, arrow: (3/2)*n*(2*n^3+15*n^2+45*n-88) end proc: seq(a(n), n = 3 .. 35);

%t Table[(3n(2n^3+15n^2+45n-88))/2,{n,3,30}] (* _Harvey P. Dale_, Mar 02 2018 *)

%Y Cf. A224456.

%K nonn

%O 3,1

%A _Emeric Deutsch_, Apr 14 2013