A238410 a(n) = floor((3(n-1)^2 + 1)/2).

0, 2, 6, 14, 24, 38, 54, 74, 96, 122, 150, 182, 216, 254, 294, 338, 384, 434, 486, 542, 600, 662, 726, 794, 864, 938, 1014, 1094, 1176, 1262, 1350, 1442, 1536, 1634, 1734, 1838, 1944, 2054, 2166, 2282, 2400, 2522, 2646, 2774, 2904, 3038, 3174, 3314, 3456, 3602, 3750, 3902, 4056, 4214, 4374, 4538, 4704

Offset: 1

Author

Emeric Deutsch, Feb 27 2014

Keywords

• nonn
• easy

Comments

a(n) = the eccentric connectivity index of the path P[n] on n vertices. The eccentric connectivity index of a simple connected graph G is defined to be the sum over all vertices i of G of the product E(i)D(i), where E(i) is the eccentricity and D(i) is the degree of vertex i. For example, a(4)=14 because the vertices of P[4] have degrees 1,2,2,1 and eccentricities 3,2,2,3; we have 1*3 + 2*2 + 2*2 + 1*3 = 14.

From Paul Curtz, Feb 23 2023: (Start)

East spoke of the hexagonal spiral using A004526 with a single 0:

.

43 42 42 41 41 40

43 28 28 27 27 26 40

44 29 17 16 16 15 26 39

44 29 17 8 8 7 15 25 39

45 30 18 9 3 2 7 14 25 38

45 30 18 9 3 0---2---6--14--24--38-->

31 19 10 4 1 1 6 13 24 37

31 19 10 4 5 5 13 23 37

32 20 11 11 12 12 23 36

32 20 21 21 22 22 36

33 33 34 34 35 35

.

Other spokes are A032528, A005449, A227017, A045943, A005448, A212959, A000326, A282513, A143689, and A104249. (End)

Links

• Matthew House, Table of n, a(n) for n = 1..10000
• M. J. Morgan, S. Mukwembi and H. C. Swart, On the eccentric connectivity index of a graph, Discrete Math., 311, 2011, 1229-1234.
• B. Zhou and Zh. Du, On eccentric connectivity index, Comm. Math. Comp. Chem. (MATCH), 63, 2010, 181-198.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A004526, A077043.

(Cf. A056105, A056107.)

Programs

Maple

a := proc (n) options operator, arrow: floor((3/2)*(n-1)^2+1/2) end proc: seq(a(n), n = 1 .. 70);

Mathematica

Table[Floor[(3(n-1)^2+1)/2],{n,80}]  (* or *) LinearRecurrence[{2,0,-2,1},{0,2,6,14},80] (* Harvey P. Dale, Apr 30 2022 *)

PARI

a(n)=(3*(n-1)^2 + 1)\2 \\ Charles R Greathouse IV, Feb 15 2017

Formula

a(n) = (3*n)^2/6 for n even and a(n) = ((3*n)^2 + 3)/6 for n odd. - Miquel Cerda, Jun 17 2016

From Ilya Gutkovskiy, Jun 17 2016: (Start)

G.f.: 2*x^2*(1 + x + x^2)/((1 - x)^3*(1 + x)).

a(n) = (6*n^2 - 12*n + 7 + (-1)^n)/4.

a(n) = 2* A077043(n-1). (End)

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Matthew House, Feb 15 2017

Sum_{n>=2} 1/a(n) = Pi^2/36 + tanh(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Mar 12 2023