OFFSET
1,2
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
.
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).
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
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 *)
PROG
(PARI) a(n)=(3*(n-1)^2 + 1)\2 \\ Charles R Greathouse IV, Feb 15 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Feb 27 2014
STATUS
approved