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The hyper-Wiener index of the nanostar dendrimer defined pictorially in Fig. 1 of the Iranmanesh et al. reference.
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%I #15 Dec 03 2023 12:26:13

%S 449908,1096118,3588202,14360018,64595362,307414082,1492075906,

%T 7247514626,34935347458,166607364866,785654642434,3665131036418,

%U 16929529343746,77501591179010,351950298746626,1586770938400514,7107632205434626,31651156830097154

%N The hyper-Wiener index of the nanostar dendrimer defined pictorially in Fig. 1 of the Iranmanesh et al. reference.

%C a(1) and a(2) have been checked by the direct computation of the hyper-Wiener index (using Maple).

%H Colin Barker, <a href="/A227702/b227702.txt">Table of n, a(n) for n = 1..1000</a>

%H A. Iranmanesh, N. A. Gholami, <a href="https://hrcak.srce.hr/28365">Computing the Szeged index of two type dendrimer nanostars</a>, Croatica Chemica Acta, 81, No. 2, 2008, 299-303.

%H T. Tada, D. Nozaki, M. Kondo, K. Yoshizawa, <a href="https://doi.org/10.1021/jp0309724">Molecular orbital interactions in the nanostar dendrimer</a>, J. Phys. Chem. B 107, 2003, 14204-14210.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (19,-150,636,-1560,2208,-1664,512).

%F a(n) = 167682 + 2^n*(79709 + 24876*n + 3996*n^(2)) + 4^n*(13458 + 1512* n + 1296*n^(2)).

%F G.f.: 2*x*(224954 - 3726067*x + 25124080*x^2 - 87769804*x^3 + 167355376*x^4 -165722176*x^5 + 66777344*x^6)/((1-x)*(1-2*x)^3*(1-4*x)^3).

%F a(n) = 19*a(n-1) - 150*a(n-2) + 636*a(n-3) - 1560*a(n-4) + 2208*a(n-5) - 1664*a(n-6) + 512*a(n-7) for n>6. - _Colin Barker_, May 30 2018

%p a:= proc (n) options operator, arrow: 167682+2^n*(79709+24876*n+3996*n^2)+4^n*(13458+1512*n+1296*n^2) end proc: seq(aa(n), n = 1 .. 20);

%t LinearRecurrence[{19,-150,636,-1560,2208,-1664,512},{449908,1096118,3588202,14360018,64595362,307414082,1492075906},20] (* _Harvey P. Dale_, Dec 03 2023 *)

%o (PARI) Vec(2*x*(224954 - 3726067*x + 25124080*x^2 - 87769804*x^3 + 167355376*x^4 - 165722176*x^5 + 66777344*x^6) / ((1 - x)*(1 - 2*x)^3*(1 - 4*x)^3) + O(x^20)) \\ _Colin Barker_, May 30 2018

%Y Cf. A227701.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, Jul 21 2013