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a(n) = 108*n^2 - 104*n + 20 (n>=1).
2

%I #19 Jul 29 2019 12:09:32

%S 24,244,680,1332,2200,3284,4584,6100,7832,9780,11944,14324,16920,

%T 19732,22760,26004,29464,33140,37032,41140,45464,50004,54760,59732,

%U 64920,70324,75944,81780,87832,94100,100584,107284,114200,121332,128680,136244,144024,152020,160232,168660,177304,186164

%N a(n) = 108*n^2 - 104*n + 20 (n>=1).

%C For n>=2, a(n) is the second Zagreb index of the (n,n)-triangular parallelogram P[n,n], defined in the Shiu et al. reference.

%C The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.

%C The M-polynomial of the (n,n)-triangular parallelogram P[n,n] is M(P[n,n]; x,y) = 4*x^2*y^4 + 4*x^3*y^4 + 2*x^3*y^6 +2*(2*n-3)*x^4*y^4 + 4*(2*n-3)*x^4*y^6 +(3*n^2 -10*n+8)*x^6*y^6.

%C More generally, the M-polynomial of the (p,q)-triangular parallelogram is M(P[p,q]; x,y) = 4*x^2*y^4 + 4*x^3*y^4 + 2*x^3*y^6 +2*(p + q - 3)*x^4*y^4 + 4*(p + q - 3)*x^4*y^6 +(3*p*q - 5*p -5*q +8)*x^6*y^6.

%C 27*a(n) + 136 is a square. - _Bruno Berselli_, May 21 2018

%H Colin Barker, <a href="/A304835/b304835.txt">Table of n, a(n) for n = 1..1000</a>

%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. C. Shiu, P. C. B. Lam, and K. K. Poon, <a href="https://doi.org/10.1016/S0166-218X(01)00317-1">On Wiener numbers of polygonal nets</a>, Discrete Appl. Math., 122, 2001, 251-261.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F From _Bruno Berselli_, May 21 2018: (Start)

%F G.f.: 4*x*(6 + 43*x + 5*x^2)/(1 - x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F a(n) = 4*A024204(9*n-5). (End)

%p seq(20-104*n+108*n^2, n = 1 .. 45);

%t Table[108 n^2 - 104 n + 20, {n, 1, 50}] (* _Bruno Berselli_, May 21 2018 *)

%t LinearRecurrence[{3,-3,1},{24,244,680},50] (* _Harvey P. Dale_, Jul 29 2019 *)

%o (GAP) List([1..50], n->108*n^2-104*n+20); # _Muniru A Asiru_, May 20 2018

%o (PARI) Vec(4*x*(6 + 43*x + 5*x^2)/(1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 23 2018

%Y Cf. A024204, A304834.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, May 20 2018