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The hyper-Wiener index of the straight pentachain of n pentagonal rings (see Fig. 2.1 in the A. A. Ali et al. reference).
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%I #15 Jun 23 2023 10:52:07

%S 91,263,589,1126,1940,3106,4708,6839,9601,13105,17471,22828,29314,

%T 37076,46270,57061,69623,84139,100801,119810,141376,165718,193064,

%U 223651,257725,295541,337363,383464,434126,489640,550306,616433,688339,766351,850805,942046,1040428,1146314,1260076

%N The hyper-Wiener index of the straight pentachain of n pentagonal rings (see Fig. 2.1 in the A. A. Ali et al. reference).

%H A. A. Ali and A. M. Ali, <a href="https://match.pmf.kg.ac.rs/electronic_versions/Match65/n3/match65n3_807-819.pdf">Hosoya polynomials of pentachains</a>, Comm. Math. Comp. Chem. (MATCH), 65, 2011, 807-819.

%H I. Gutman, W. Yan, Y.-N. Yeh, and B.-Y. Yang, <a href="https://www.math.sinica.edu.tw/www/file_upload/mayeh/2007zigzagging.pdf">Generalized Wiener indices of zigzagging pentachains</a>, J. Math. Chem., 42, 2007, 103-117.

%H O. Halakoo, O. Khormali, and A. Mahmiani, <a href="https://www.chalcogen.ro/687_Halkao-Khormali-sept14.pdf">Bounds for Schultz index of pentachains</a>, Digest J. Nanomaterials and Biostructures, 4, 2009, 687-691.

%H N. P. Rao and A. L. Prasanna, <a href="http://m-hikari.com/ams/ams-password-2008/ams-password49-52-2008/rao-prasannaAMS49-52-2008.pdf">On the Wiener index of pentachains</a>, Applied Math. Sci., 2, 2008, 2443-2457.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = (3*n^4 +34*n^3 +145*n^2 -190*n +208)/8.

%F G.f.: z^2*(91-192*z+184*z^2-99*z^3+25*z^4)/(1-z)^5.

%F The Hosoya polynomial is [t - 4t^2 - 3t^3 - 2t^5 - 3t^6 + 2t^7 + 4nt - nt^2 - 3nt^3 + nt^5 - nt^7 + t^{n+2} + 4t^{n+3} + 4t^{n+4}](t-1)^2.

%p a := proc (n) options operator, arrow: (3/8)*n^4+(17/4)*n^3+(145/8)*n^2-(95/4)*n+26 end proc: seq(a(n), n = 2 .. 40);

%Y Cf. A224459.

%K nonn,easy

%O 2,1

%A _Emeric Deutsch_, Jun 29 2013