|
%I #21 Feb 13 2024 15:46:40
%S 16,244,892,2764,8236,24364,72172,214444,638956,1907884,5705452,
%T 17079724,51165676,153349804,459754732,1378674604,4134844396,
%U 12402174124,37201804012,111595975084,334769051116,1004269404844,3012732717292,9038047157164,27113839481836,81340914465964,244021535438572,732062190396844
%N a(n) = 32*3^n + 18*2^n - 116 (n>=1).
%C For n>=2, a(n) is the second Zagreb index of the Sierpinski Gasket Rhombus graph SR[n] (see the Antony Xavier 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 Sierpinski Gasket Rhombus graph SR[n] is M(SR[n]; x,y) = 4*x^2*y^4 + 4*x^3*y^4 +2*x^3*y^6 + (2*3^n - 3*2^n - 4)*x^4*y^4 + (2^{n+1} - 4)*x^4*y^6 + (2^{n-1} - 2)*x^6*y^6.
%H Colin Barker, <a href="/A304170/b304170.txt">Table of n, a(n) for n = 0..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 D. Antony Xavier, M. Rosary, and Andrew Arokiaraj, <a href="https://www.ijmsc.com/index.php/ijmsc/article/view/261">Topological properties of Sierpinski Gasket Rhombus graphs</a>, International J. of Mathematics and Soft Computing, 4, No. 2, 2014, 95-104.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F From _Colin Barker_, May 12 2018: (Start)
%F G.f.: 4*(4 + 37*x - 99*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
%F a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>2.
%F (End)
%p seq(32*3^n+18*2^n-116, n = 1 .. 40);
%t CoefficientList[Series[4*(4 + 37*x - 99*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Jan 20 2024 *)
%t LinearRecurrence[{6,-11,6},{16,244,892},30] (* _Harvey P. Dale_, Feb 13 2024 *)
%o (PARI) Vec(4*(4 + 37*x - 99*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ _Colin Barker_, May 12 2018
%Y Cf. A304169.
%K nonn,easy
%O 0,1
%A _Emeric Deutsch_, May 11 2018
|
