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A245969 The average Wiener index of the set of all fibonacenes with n hexagons. 1

%I

%S 271,537,931,1477,2199,3121,4267,5661,7327,9289,11571,14197,17191,

%T 20577,24379,28621,33327,38521,44227,50469,57271,64657,72651,81277,

%U 90559,100521,111187,122581,134727,147649,161371,175917,191311,207577,224739,242821,261847,281841

%N The average Wiener index of the set of all fibonacenes with n hexagons.

%C For the definition of a fibonacene see the Gutman-Klavzar reference.

%C The number of fibonacenes with n hexagons is A005418(n-2).

%H Vincenzo Librandi, <a href="/A245969/b245969.txt">Table of n, a(n) for n = 3..1000</a>

%H A. A. Dobrynin, I. Gutman, <a href="https://doi.org/10.1016/S0097-8485(99)00035-2">The average Wiener index of hexagonal chains</a>, Computers & Chemistry, 23, 1999, 571-576.

%H I. Gutman, S. Klav┼żar, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match55/n1/match55n1_39-54.pdf">Chemical graph theory of fibonacenes</a>, Commun. Math. Chem. (MATCH), 55, 2006, 39-54.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = 4n^3 + 16n^2 + 6n + 1 (n>=3) (see p. 45 of the Gutman-Klavzar reference).

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Colin Barker_, Aug 31 2014

%F G.f.: -x^3*(109*x^3-409*x^2+547*x-271) / (x-1)^4. - _Colin Barker_, Aug 31 2014

%e a(5)=931. Indeed, there are A005418(3)=3 fibonacenes with 5 hexagons (see Fig. 1 of the Gutman-Klavzar reference); their Wiener indices are 963, 931, and 899; average is 931.

%p seq(4*n^3+16*n^2+6*n+1, n = 3 .. 40);

%t Table[4 n^3 + 16 n^2 + 6 n + 1, {n, 3, 40}] (* _Vincenzo Librandi_, Aug 31 2014 *)

%o (PARI) Vec(-x^3*(109*x^3-409*x^2+547*x-271)/(x-1)^4 + O(x^100)) \\ _Colin Barker_, Aug 31 2014

%o (MAGMA) [4*n^3 + 16*n^2 + 6*n + 1: n in [3..50]]; // _Vincenzo Librandi_, Aug 31 2014

%Y Cf. A005418.

%K nonn,easy

%O 3,1

%A _Emeric Deutsch_, Aug 17 2014

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Last modified July 25 10:18 EDT 2021. Contains 346289 sequences. (Running on oeis4.)