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).

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

Table of n, a(n) for n=1..40.

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

Cf. A143938, A033544.

Sequence in context: A129026 A042426 A042424 * A193391 A193399 A193393

Adjacent sequences: A248092 A248093 A248094 * A248096 A248097 A248098

Keyword

nonn,tabl

Author

Emeric Deutsch, Nov 15 2014

Status

approved