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a(n) = 36*n^2 - 8*n - 2 (n >=1).
2

%I #20 Sep 21 2018 11:10:04

%S 26,126,298,542,858,1246,1706,2238,2842,3518,4266,5086,5978,6942,7978,

%T 9086,10266,11518,12842,14238,15706,17246,18858,20542,22298,24126,

%U 26026,27998,30042,32158,34346,36606,38938,41342,43818,46366,48986,51678,54442,57278,60186,63166,66218,69342,72538

%N a(n) = 36*n^2 - 8*n - 2 (n >=1).

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

%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 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 Sequence found by reading the line from 26, in the direction 26, 126, ..., in the square spiral whose vertices are the generalized 20-gonal numbers. - _Omar E. Pol_, May 20 2018

%C 9*a(n) + 22 is a square. - _Bruno Berselli_, May 21 2018

%H Colin Barker, <a href="/A304834/b304834.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 G.f. 2*x*(13 + 24*x - x^2)/(1 - x)^3. - _Bruno Berselli_, May 21 2018

%p seq(-2-8*n+36*n^2, n = 1 .. 45);

%t Table[36 n^2 - 8 n - 2, {n, 1, 50}] (* _Bruno Berselli_, May 21 2018 *)

%t LinearRecurrence[{3,-3,1},{26,126,298},50] (* _Harvey P. Dale_, Sep 21 2018 *)

%o (GAP) List([1..50], n->36*n^2-8*n-2); # _Muniru A Asiru_, May 20 2018

%o (PARI) Vec(2*x*(13 + 24*x - x^2)/(1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 23 2018

%Y Cf. A304380, A304835.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, May 20 2018