login
A248095
Triangle read by rows: TR(m,n) is the Wiener index of the hexagonal trapezium T(m,n), defined in the He et al. reference (1 <= n <= m).
0
27, 109, 210, 279, 566, 822, 569, 1182, 1816, 2328, 1011, 2130, 3370, 4540, 5433, 1637, 3482, 5612, 7772, 9707, 11130, 2479, 5310, 8670, 12224, 15653, 18622, 20748, 3569, 7686, 12672, 18096, 23559, 28662, 32974, 36000, 4939, 10682, 17746, 25588
OFFSET
1,1
COMMENTS
m denotes the number of hexagons in the bottom row, while n is the number of rows of hexagons.
TR(m,1) = A143938(m) = Wiener index of a linear chain of m hexagons.
TR(n,n) = A033544(n) = Wiener index of an n-hexagonal triangle.
LINKS
Q. H. He, J. Z. Gu, S. J. Xu, and W. H. Chan, Hosoya polynomials of hexagonal triangles and trapeziums, MATCH, Commun. Math. Comput. Chem. 72, 2014, 835-843.
FORMULA
TR(m,n) = (4*m^3*(n + 1)^2 + 2*m^2*(3 + 11*n + 6*n^2 - 2*n^3))/3 + ((n*(28 + 45*n - 35*n^2 - 8*n^4)+20*m*(1 + 9*n + 6*n^2 - 4*n^3 + n^4))/30; see Corollary 3,7 in the He et al. reference.
The reader can get the lengthy expression of the bivariate g.f. G by activating the Maple program for TR(m,n) and then activating the Maple program for G.
EXAMPLE
Row 1 is 27; indeed T(1,1) is just one hexagon ABCDEF; it has 6 distances equal to 1 (= number of edges), 6 distances equal to 2 (AC, BD, CE, DA, EA, FB), and 3 distances equal to 3 (AD, BE, CF); 6*1 + 6*2 + 3*3 = 27.
Triangle starts:
27;
109, 210;
279, 566, 822;
569, 1182, 1816, 2328;
MAPLE
TR := proc (m, n) options operator, arrow: (4/3)*m^3*(n+1)^2+(2/3)*m^2*(3+11*n+6*n^2-2*n^3)+(1/30)*n*(28+45*n-35*n^2-8*n^4)+(2/3)*m*(1+9*n+6*n^2-4*n^3+n^4) end proc: for m to 10 do seq(TR(m, n), n = 1 .. m) end do; # yields sequence in triangular form
G := factor(sum(sum(TR(i, j)*z^i*t^j, j = 1 .. i), i = 1 .. infinity));
PROG
(Magma) /* As triangle */ [[(4*m^3*(n + 1)^2 + 2*m^2*(3 + 11*n + 6*n^2 - 2*n^3))/3 + ((n*(28 + 45*n - 35*n^2 - 8*n^4)+20*m*(1 + 9*n + 6*n^2 - 4*n^3 + n^4)) / 30): n in [1..m]]: m in [1.. 15]]; // Vincenzo Librandi, Nov 16 2014
CROSSREFS
Sequence in context: A129026 A042426 A042424 * A193391 A193399 A193393
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Nov 15 2014
STATUS
approved