A248093
Triangle read by rows: TR(n,k) is the number of unordered vertex pairs at distance k of the hexagonal triangle T_n, defined in the He et al. reference (1<=k<=2n+1).
Original entry on oeis.org
1, 0, 6, 6, 6, 3, 13, 15, 21, 21, 15, 6, 22, 27, 42, 48, 45, 36, 24, 9, 33, 42, 69, 84, 87, 81, 69, 51, 33, 12, 46, 60, 102, 129, 141, 141, 132, 114, 93, 66, 42, 15, 61, 81, 141, 183, 207, 216, 213, 198, 177, 147, 117, 81, 51, 18, 78, 105, 186, 246, 285
Offset: 0
Row n=1 is 6, 6, 6, 3; indeed, T_1 is a hexagon ABCDEF; it has 6 distances equal to 0 (= number of vertices), 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).
Triangle starts:
1, 0;
6, 6, 6, 3;
13, 15, 21, 21, 15, 6;
22, 27, 42, 48, 45, 36, 24, 9;
33, 42, 69, 84, 87, 81, 69, 51, 33, 12;
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G := (1+(3+6*t+4*t^2+3*t^3)*z-(1+t+2*t^2)*(2+t-2*t^2)*z^2+t^2*(1-3*t^2)*z^3+t^4*z^4)/((1-z)^3*(1-t^2*z)^2): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 22 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. 2*n+1) end do; # yields sequence in triangular form
A248094
The hyper-Wiener index of the hexagonal triangle T_n, defined in the He et al. reference.
Original entry on oeis.org
0, 42, 444, 2187, 7443, 20247, 47313, 98994, 190386, 342576, 584034, 952149, 1494909, 2272725, 3360399, 4849236, 6849300, 9491814, 12931704, 17350287, 22958103, 29997891, 38747709, 49524198, 62685990, 78637260, 97831422, 120774969, 148031457, 180225633, 218047707
Offset: 0
- 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.
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
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[n*(66+407*n+670*n^2+425*n^3+104*n^4+8*n^5)/40: n in [0..30]]; // Vincenzo Librandi, Nov 15 2014
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a := n -> (1/40)*n*(66 + 407*n + 670*n^2 + 425*n^3 + 104*n^4 + 8*n^5): seq(a(n), n = 0 .. 30);
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CoefficientList[Series[x (42 + 150 x - 39 x^2 - 12 x^3 + 3 x^4) / (1 - x)^7, {x, 0, 30}], x] (* Vincenzo Librandi, Nov 15 2014 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,42,444,2187,7443,20247,47313},40] (* Harvey P. Dale, Oct 22 2022 *)
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).
Original entry on oeis.org
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
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;
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/* 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
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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));
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