|
%I #19 Jun 23 2023 10:52:13
%S 55,133,259,442,691,1015,1423,1924,2527,3241,4075,5038,6139,7387,8791,
%T 10360,12103,14029,16147,18466,20995,23743,26719,29932,33391,37105,
%U 41083,45334,49867,54691,59815,65248,70999,77077,83491,90250,97363,104839,112687
%N The 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_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = (3*n^3 +21*n^2 -6*n +14)/2.
%F G.f.: z^2*(55-87*z+57*z^2-16*z^3)/(1-z)^4.
%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: (3/2)*n^3+(21/2)*n^2-3*n+7 end proc: seq(a(n), n = 2 .. 40);
%t LinearRecurrence[{4,-6,4,-1},{55,133,259,442},40] (* _Harvey P. Dale_, Mar 18 2023 *)
%Y Cf. A224460.
%K nonn,easy
%O 2,1
%A _Emeric Deutsch_, Jun 29 2013
|
