A224454 -id:A224454 - cp's OEIS frontend

A224455 The hyper-Wiener index of the linear phenylene with n hexagons.

42, 396, 1656, 4740, 10890, 21672, 38976, 65016, 102330, 153780, 222552, 312156, 426426, 569520, 745920, 960432, 1218186, 1524636, 1885560, 2307060, 2795562, 3357816, 4000896, 4732200, 5559450, 6490692, 7534296, 8698956, 9993690, 11427840, 13011072, 14753376, 16665066, 18756780, 21039480

Offset: 1

Emeric Deutsch, Apr 10 2013

a(2) and a(5) have been checked by the direct computation of the hyper-Wiener index (using Maple).

I. Gutman, The topological indices of linear phenylenes, J. Serb. Chem. Soc., 60, No. 2, 1995, 99-104.

G. C. Greubel, Table of n, a(n) for n = 1..1000
G. Cash, S. Klavzar, M. Petkovsek, Three methods for calculation of the hyper-Wiener index of a molecular graph, J. Chem. Inf. Comput. Sci. 42, 2002, 571-576.
L. Pavlovic, I. Gutman, Wiener numbers of phenylenes: an exact result, J. Chem. Inf. Comput. Sci. 37, 1997, 355-358.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A224454.

a := proc (n) options operator, arrow: (3/2)*n*(n+1)*(9*n^2+3*n+2) end proc: seq(a(n), n = 1 .. 35);

LinearRecurrence[{5,-10,10,-5,1}, {42,396,1656,4740,10890}, 100] (* or *) Table[(3/2)*n*(n+1)*(9*n^2 + 3*n + 2), {n,1,100}] (* G. C. Greubel, Dec 08 2016 *)

Vec(6*x*(7 + 31*x + 16*x^2)/(1-x)^5 + O(x^50)) \\ G. C. Greubel, Dec 08 2016

a(n) = (3/2)*n*(n+1)*(9*n^2 + 3*n + 2).
G.f.: 6*x*(7 + 31*x + 16*x^2)/(1-x)^5.
The Hosoya polynomial of the linear phenylene with n hexagons is nt(t^3-t^2-4t-8)/(t-1) + 2t(t+1)(t^(3n)-1)/(t-1)^2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - G. C. Greubel, Dec 08 2016

A304157 a(n) is the first Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference.

Original entry on oeis.org

24, 68, 112, 156, 200, 244, 288, 332, 376, 420, 464, 508, 552, 596, 640, 684, 728, 772, 816, 860, 904, 948, 992, 1036, 1080, 1124, 1168, 1212, 1256, 1300, 1344, 1388, 1432, 1476, 1520, 1564, 1608, 1652, 1696, 1740, 1784, 1828, 1872, 1916, 1960, 2004, 2048

Offset: 1

Emeric Deutsch, May 07 2018

The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of the linear phenylene G[n] is M(G[n];x,y) = 6*x^2*y^2 + 4*(n - 1)*x^2*y^3 + 4(n - 1)*x^3*y^3.
a(n) is the first Zagreb index of the angular phenylene shown in the Bodroza-Pantic et al. reference (Fig. 1 (b)). - Emeric Deutsch, May 24 2018

			From _Andrew Howroyd_, May 09 2018: (Start)
Illustration of the first two graphs:
       o              o         o
     /   \          /   \     /   \
    o     o        o     o---o     o
    |     |        |     |   |     |
    o     o        o     o---o     o
     \   /          \   /     \   /
       o              o         o
In general, the graph consists of a chain of n linked hexagons.
.
Case n=1: There are 6 vertices of degree 2, so a(1) = 6*2^2 = 24.
Case n=2: There are 8 vertices of degree 2 and 4 of degree 3, so a(2) = 8*2^2 + 4*3^3 = 32 + 36 = 68.
In general, there will be 2n + 4 vertices of degree 2 and 4n - 4 of degree 3.
(End)

Colin Barker, Table of n, a(n) for n = 1..1000
O. Bodroza-Pantic, I. Gutman, and S. J. Cyvin, Fibonacci numbers and algebraic structure count of some non-benzenoid conjugated polymers, The Fibonacci Quarterly, 35, 1, 1997, 75-83.
M. R. Darafsheh, Computation of topological indices of some graphs, Acta Appl. Math., 110, 2010, 1225-1235.
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
P. Gayathri and U. Priyanka, Degree based topological indices of linear phenylene, Internat. J. of Innovative Research in Science, Engineering and Technology,6, 8, 2017, 16986-16997.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017461, A224454, A224455, A304158.

Maple
```
seq(44*n - 20, n = 1 .. 40);
```

Vec(4*x*(6 + 5*x) / (1 - x)^2 + O(x^60)) \\ Colin Barker, May 07 2018

a(n) = 44*n - 20.
a(n) = 4 * A017461(n-1).
From Colin Barker, May 07 2018: (Start)
G.f.: 4*x*(6 + 5*x) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>2.
(End)

cp's OEIS Frontend

A224455 The hyper-Wiener index of the linear phenylene with n hexagons.

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