OFFSET
1,1
COMMENTS
a(n) is the first Zagreb index of the (n,n)-triangular parallelogram P[n,n], defined in the Shiu et al. reference.
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums 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.
Sequence found by reading the line from 26, in the direction 26, 126, ..., in the square spiral whose vertices are the generalized 20-gonal numbers. - Omar E. Pol, May 20 2018
9*a(n) + 22 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
G.f. 2*x*(13 + 24*x - x^2)/(1 - x)^3. - Bruno Berselli, May 21 2018
MAPLE
seq(-2-8*n+36*n^2, n = 1 .. 45);
MATHEMATICA
Table[36 n^2 - 8 n - 2, {n, 1, 50}] (* Bruno Berselli, May 21 2018 *)
LinearRecurrence[{3, -3, 1}, {26, 126, 298}, 50] (* Harvey P. Dale, Sep 21 2018 *)
PROG
(GAP) List([1..50], n->36*n^2-8*n-2); # Muniru A Asiru, May 20 2018
(PARI) Vec(2*x*(13 + 24*x - 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