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a(n) = n*(n+17).
19

%I #72 Jan 16 2021 04:20:23

%S 0,18,38,60,84,110,138,168,200,234,270,308,348,390,434,480,528,578,

%T 630,684,740,798,858,920,984,1050,1118,1188,1260,1334,1410,1488,1568,

%U 1650,1734,1820,1908,1998,2090,2184,2280,2378,2478,2580,2684,2790,2898,3008,3120

%N a(n) = n*(n+17).

%C a(n) is the first Zagreb index of the helm graph H[n] (n>=3). - _Emeric Deutsch_, Nov 05 2016

%C From _Emeric Deutsch_, Nov 07 2016: (Start)

%C a(n) is the first Zagreb index of the graph obtained by joining one vertex of the cycle graph C[n] with each vertex of a second cycle graph C[n].

%C The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. (End)

%C From _Emeric Deutsch_, May 11 2018: (Start)

%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.

%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. (End)

%H Muniru A Asiru, <a href="/A132761/b132761.txt">Table of n, a(n) for n = 0..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 Ivan Gutman and Kinkar Ch. Das, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf">The first Zagreb index 30 years after</a>, MATCH Commun. Math. Comput. Chem. 50 (2004), pp. 83-92.

%H Felix P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, Preprint on ResearchGate, March 2014.

%H Eric Weisstein's World of Mathematics, <a href="http://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 a(n) = n*(n + 17).

%F a(n) = A132760(n) + 2*n = A132765(n) - 6*n = A098849(n) + 1*n = A120071(n) - 3*n. - _Zerinvary Lajos_, Feb 17 2008

%F a(n) = 2*n + a(n-1) + 16 for n>0, a(0)=0. - _Vincenzo Librandi_, Aug 03 2010

%F G.f.: 2*x*(9 - 8*x)/(1 - x)^3. - _Emeric Deutsch_, Nov 07 2016

%F From _Amiram Eldar_, Jan 16 2021: (Start)

%F Sum_{n>=1} 1/a(n) = H(17)/17 = A001008(17)/A102928(17) = 42142223/208288080, where H(k) is the k-th harmonic number.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/17 - 1768477/41657616. (End)

%t Table[n(n+17),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,18,38},50] (* _Harvey P. Dale_, Sep 12 2020 *)

%o (PARI) a(n)=n*(n+17) \\ _Charles R Greathouse IV_, Nov 07 2016

%o (GAP) List([0..50],n->n*(n+17)); # _Muniru A Asiru_, May 11 2018

%Y Cf. A001008, A002378, A098849, A102928, A120071, A132760, A132765.

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Aug 28 2007