OFFSET
1,1
COMMENTS
For n>=2, a(n) is the second Zagreb index of the (n,n)-triangular parallelogram P[n,n], defined in the Shiu et al. reference.
The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.
The M-polynomial of the (n,n)-triangular parallelogram P[n,n] is M(P[n,n]; x,y) = 4*x^2*y^4 + 4*x^3*y^4 + 2*x^3*y^6 +2*(2*n-3)*x^4*y^4 + 4*(2*n-3)*x^4*y^6 +(3*n^2 -10*n+8)*x^6*y^6.
More generally, the M-polynomial of the (p,q)-triangular parallelogram is M(P[p,q]; x,y) = 4*x^2*y^4 + 4*x^3*y^4 + 2*x^3*y^6 +2*(p + q - 3)*x^4*y^4 + 4*(p + q - 3)*x^4*y^6 +(3*p*q - 5*p -5*q +8)*x^6*y^6.
27*a(n) + 136 is a square. - Bruno Berselli, May 21 2018
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
W. C. Shiu, P. C. B. Lam, and K. K. Poon, On Wiener numbers of polygonal nets, Discrete Appl. Math., 122, 2001, 251-261.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From Bruno Berselli, May 21 2018: (Start)
G.f.: 4*x*(6 + 43*x + 5*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 4*A024204(9*n-5). (End)
MAPLE
seq(20-104*n+108*n^2, n = 1 .. 45);
MATHEMATICA
Table[108 n^2 - 104 n + 20, {n, 1, 50}] (* Bruno Berselli, May 21 2018 *)
LinearRecurrence[{3, -3, 1}, {24, 244, 680}, 50] (* Harvey P. Dale, Jul 29 2019 *)
PROG
(GAP) List([1..50], n->108*n^2-104*n+20); # Muniru A Asiru, May 20 2018
(PARI) Vec(4*x*(6 + 43*x + 5*x^2)/(1 - x)^3 + O(x^40)) \\ Colin Barker, May 23 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, May 20 2018
STATUS
approved