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a(n) = 4*n*(n+5).
1

%I #41 Feb 16 2025 08:33:36

%S 24,56,96,144,200,264,336,416,504,600,704,816,936,1064,1200,1344,1496,

%T 1656,1824,2000,2184,2376,2576,2784,3000,3224,3456,3696,3944,4200,

%U 4464,4736,5016,5304,5600,5904,6216,6536,6864,7200,7544,7896,8256,8624,9000,9384,9776

%N a(n) = 4*n*(n+5).

%C a(n) is the second Zagreb index of the helm graph H[n] (n>=3).

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

%C The M-polynomial of the Helm graph H[n] is M(H[n]; x,y) = n*x*y^4 + n*x^4*y^4 + n*x^4*y^n. - _Emeric Deutsch_, May 11 2018

%C The helm graph H[n] is the graph obtained from an n-wheel graph by adjoining a pendant edge at each node of the cycle. - _Emeric Deutsch_, May 11 2018

%C a(n) - 16*n + 1 is a square. - _Muniru A Asiru_, Jun 01 2018

%H Muniru A Asiru, <a href="/A277108/b277108.txt">Table of n, a(n) for n = 1..5000</a>

%H Emeric 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, Vol. 6, No. 2, 2015, pp. 93-102.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HelmGraph.html">Helm Graph</a>.

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

%F G.f.: 8*z*(3-2*z)/(1-z)^3.

%F a(n) = 4*A028557(n) = 8*A055998(n).

%F From _Elmo R. Oliveira_, Jan 28 2025: (Start)

%F E.g.f.: 4*exp(x)*x*(6 + x).

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

%p seq(4*n^2+20*n, n = 1 .. 40);

%t Table[4 n (n + 5), {n, 40}] (* or *)

%t Rest@ CoefficientList[Series[8 x (3 - 2 x)/(1 - x)^3, {x, 0, 40}], x] (* _Michael De Vlieger_, Nov 06 2016 *)

%o (PARI) a(n)=4*n*(n+5) \\ _Charles R Greathouse IV_, Jun 17 2017

%o (GAP) List([1..50],n->4*n*(n+5)); # _Muniru A Asiru_, Jun 01 2018

%Y Cf. A028557, A055998, A132761.

%K nonn,easy,changed

%O 1,1

%A _Emeric Deutsch_, Nov 05 2016