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%I #22 Sep 08 2022 08:46:17
%S -30,-8,38,108,202,320,462,628,818,1032,1270,1532,1818,2128,2462,2820,
%T 3202,3608,4038,4492,4970,5472,5998,6548,7122,7720,8342,8988,9658,
%U 10352,11070,11812,12578,13368,14182,15020,15882,16768,17678,18612,19570
%N a(n) = 12*n^2 + 10*n - 30.
%C For n>=3, a(n) is the second Zagreb index of the uniform bow graph B[n]. 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. The uniform bow graph B[n] consists of two path graphs P[n] and an additional vertex joined by 2n edges to the vertices of the paths.
%C The M-polynomial of the uniform bow graph B[n] is M(B[n],x,y) = 4*x^2*y^3 + 4*x^2*y^{2*n} + (2*n-6)*x^3*y^3 + (2*n-4)*x^3*y^{2*n}.
%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 J. Jeba Jesintha and K. Ezhilarasi Hilda, <a href="http://dx.doi.org/10.1007/s11786-015-0224-2">All uniform bow graphs are graceful</a>, Math. Comput. Sci., 9, 2015, 185-191.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F O.g.f.: 2*(7*x - 3)*(2*x - 5)/(x - 1)^3.
%F E.g.f.: 2*(6*x^2 + 11*x - 15)*exp(x). - _Bruno Berselli_, Nov 11 2016
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Nov 11 2016
%p seq(12*n^2+10*n-30, n=0..40);
%t Table[12 n^2 + 10 n - 30, {n, 0, 50}] (* _Vincenzo Librandi_, Nov 11 2016 *)
%t LinearRecurrence[{3,-3,1},{-30,-8,38},50] (* _Harvey P. Dale_, Apr 19 2020 *)
%o (Sage) [12*n^2+10*n-30 for n in range(50)] # _Bruno Berselli_, Nov 11 2016
%o (Magma) [12*n^2+10*n-30: n in [0..50]]; // _Vincenzo Librandi_, Nov 11 2016
%o (PARI) a(n)=12*n^2+10*n-30 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A277981.
%K sign,easy
%O 0,1
%A _Emeric Deutsch_, Nov 10 2016
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