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a(n) = 344*7^n/21 - 128/3 (n>=1).
2

%I #20 Aug 24 2019 20:20:00

%S 72,760,5576,39288,275272,1927160,13490376,94432888,661030472,

%T 4627213560,32390495176,226733466488,1587134265672,11109939859960,

%U 77769579019976,544387053140088,3810709371980872,26674965603866360,186724759227064776,1307073314589453688,9149513202126176072

%N a(n) = 344*7^n/21 - 128/3 (n>=1).

%C a(n) is the first Zagreb index of the crystal structure cubic carbon CCC(n), defined in the Baig et al. and in the Gao et al. references.

%C 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.

%C For n>=2 the M-polynomial of the crystal structure cubic carbon CCC(n) is M(CCC(n); x,y) = 72*7^(n-2)*x^3*y^3 + 24*7^(n-2)*x^3*y^4 + (76*7^(n-2) - 16)*x^4*y^4/3.

%H Colin Barker, <a href="/A304828/b304828.txt">Table of n, a(n) for n = 1..1000</a>

%H A. Q. Baig, M. Imran, W. Khalid, and M. Naeem, <a href="https://doi.org/10.1139/cjc-2017-0083">Molecular description of carbon graphite and crystal cubic carbon structures</a>, Canadian J. Chem., 95, 674-686, 2017.

%H E. Deutsch and Sandi Klavzar, <a href="http://dx.doi.org/10.22052/ijmc.2015.10106">M-polynomial and degree-based topological indices</a>, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.

%H W. Gao, M. K. Siddiqui, M. Naeem and N. A. Rehman, <a href="https://doi.org/10.3390/molecules22091496">Topological characterization of carbon graphite and crystal cubic carbon structures</a>, Molecules, 22, 1496, 1-12, 2017.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-7).

%F From _Colin Barker_, May 21 2018: (Start)

%F G.f.: 8*x*(9 + 23*x) / ((1 - x)*(1 - 7*x)).

%F a(n) = 8*a(n-1) - 7*a(n-2) for n>2. (End)

%p seq((344*7^(n-1)-128)*(1/3), n = 1 .. 25);

%t LinearRecurrence[{8,-7},{72,760},30] (* _Harvey P. Dale_, Aug 24 2019 *)

%o (PARI) Vec(8*x*(9 + 23*x) / ((1 - x)*(1 - 7*x)) + O(x^30)) \\ _Colin Barker_, May 21 2018

%Y Cf. A304826, A304827, A304829.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, May 21 2018