A248095 - OEIS

%I #17 Nov 11 2024 01:57:05

%S 27,109,210,279,566,822,569,1182,1816,2328,1011,2130,3370,4540,5433,

%T 1637,3482,5612,7772,9707,11130,2479,5310,8670,12224,15653,18622,

%U 20748,3569,7686,12672,18096,23559,28662,32974,36000,4939,10682,17746,25588

%N 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).

%C m denotes the number of hexagons in the bottom row, while n is the number of rows of hexagons.

%C TR(m,1) = A143938(m) = Wiener index of a linear chain of m hexagons.

%C TR(n,n) = A033544(n) = Wiener index of an n-hexagonal triangle.

%H Q. H. He, J. Z. Gu, S. J. Xu, and W. H. Chan, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match72/n3/match72n3_835-843.pdf">Hosoya polynomials of hexagonal triangles and trapeziums</a>, MATCH, Commun. Math. Comput. Chem. 72, 2014, 835-843.

%F 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.

%F 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.

%e 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.

%e Triangle starts:

%e   27;

%e   109, 210;

%e   279, 566, 822;

%e   569, 1182, 1816, 2328;

%p 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

%p G := factor(sum(sum(TR(i, j)*z^i*t^j, j = 1 .. i), i = 1 .. infinity));

%o (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

%Y Cf. A143938, A033544.

%K nonn,tabl

%O 1,1

%A _Emeric Deutsch_, Nov 15 2014