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A248093
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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).
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1
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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
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OFFSET
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0,3
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COMMENTS
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Number of entries in row n is 2*n+2.
The entries in row n are the coefficients of the Hosoya polynomial of T_n.
TR(n,0) = A028872(n+2) = number of vertices of T_n.
TR(n,1) = A140091(n) = number of edges of T_n.
sum(j*TR(n,j), j=0..2n+1) = A033544(n) = the Wiener index of T_n.
(1/2)*sum(j*(j+1)TR(n,j), j=0..2n+1) = A248094(n) = the hyper-Wiener index of T_n.
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LINKS
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FORMULA
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G.f.: (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)^2); follows from Theorem 3.6 of the He et al. reference.
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EXAMPLE
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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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MAPLE
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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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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