A132761 a(n) = n*(n+17).

0, 18, 38, 60, 84, 110, 138, 168, 200, 234, 270, 308, 348, 390, 434, 480, 528, 578, 630, 684, 740, 798, 858, 920, 984, 1050, 1118, 1188, 1260, 1334, 1410, 1488, 1568, 1650, 1734, 1820, 1908, 1998, 2090, 2184, 2280, 2378, 2478, 2580, 2684, 2790, 2898, 3008, 3120

Offset: 0

Author

Omar E. Pol, Aug 28 2007

Keywords

• nonn
• easy

Comments

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

From Emeric Deutsch, Nov 07 2016: (Start)

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].

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)

From Emeric Deutsch, May 11 2018: (Start)

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.

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)

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..5000
• Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2 (2015), pp. 93-102.
• Ivan Gutman and Kinkar Ch. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem., No. 50 (2004), pp. 83-92.
• Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
• Eric Weisstein's World of Mathematics, Helm Graph.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A001008, A002378, A056126, A098849, A102928, A120071, A132760, A132765.

Programs

GAP

List([0..50],n->n*(n+17)); # Muniru A Asiru, May 11 2018

Mathematica

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

PARI

a(n)=n*(n+17) \\ Charles R Greathouse IV, Nov 07 2016

Formula

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

a(n) = A132760(n) + 2*n = A132765(n) - 6*n = A098849(n) + n = A120071(n) - 3*n. - Zerinvary Lajos, Feb 17 2008

a(n) = 2*n + a(n-1) + 16 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 03 2010

G.f.: 2*x*(9 - 8*x)/(1 - x)^3. - Emeric Deutsch, Nov 07 2016

From Amiram Eldar, Jan 16 2021: (Start)

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

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

From Elmo R. Oliveira, Dec 12 2024: (Start)

E.g.f.: exp(x)*x*(18 + x).

a(n) = 2*A056126(n).

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