A132759 a(n) = n*(n+13).

0, 14, 30, 48, 68, 90, 114, 140, 168, 198, 230, 264, 300, 338, 378, 420, 464, 510, 558, 608, 660, 714, 770, 828, 888, 950, 1014, 1080, 1148, 1218, 1290, 1364, 1440, 1518, 1598, 1680, 1764, 1850, 1938, 2028, 2120, 2214, 2310, 2408

Offset: 0

Author

Omar E. Pol, Aug 28 2007

Keywords

• easy
• nonn

Comments

a(n) is the first Zagreb index of the gear graph g[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. The gear graph g[n] is defined as a wheel graph with n+1 vertices with a vertex added between each pair of adjacent vertices of the outer cycle. - Emeric Deutsch, Nov 09 2016

Links

• 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, Gear Graph.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002378, A056119, A120071.

Programs

Mathematica

s=0;lst={s};Do[s+=n++ +14;AppendTo[lst, s], {n, 0, 7!, 2}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2008 *)
Table[n(n+13),{n,0,50}] (* Harvey P. Dale, Aug 22 2019 *)

PARI

a(n)=n*(n+13) \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = n*(n + 13) = 2*A056119(n).

a(n) = 2*n + a(n-1) + 12 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010

G.f.: 2*x*(-7+6*x)/(x-1)^3. - R. J. Mathar, Jul 14 2012

Sum_{n>=1} 1/a(n) = 1145993/4684680 = 0.2446256... - R. J. Mathar, Jul 14 2012

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/13 - 263111/4684680. - Amiram Eldar, Jan 15 2021

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

E.g.f.: exp(x)*x*(14 + x).

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