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A228314
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Triangle read by rows: T(p,q) (1<=q<=p) is the hyper-Wiener index of the Cartesian product of the cycles C(p) and C(q) (Torus Grid Graph).
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2
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0, 1, 10, 3, 27, 72, 10, 72, 186, 448, 20, 135, 345, 810, 1450, 42, 258, 648, 1464, 2580, 4482, 70, 413, 1029, 2282, 3990, 6846, 10388, 120, 672, 1656, 3584, 6200, 10464, 15736, 23552, 180, 981, 2403, 5130, 8820, 14742, 22050, 32760, 45360, 275, 1450, 3525, 7400, 12625
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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T(p,q) = pq(3-4pq-6p-6q-3p^2-3q^2+6pq^2+6p^2q+2pq^3+3p^2q^2+2p^3q)/96 if both p and q are odd.
T(p,q) = p^2*q^2*(8+6p + 6q+2p^2+3pq+2q^2)/96 if both p and q are even.
T(p,q) = pq^2*(2p-3q-6+6pq+ 6p^2 +2pq^2+3p^2q+2p^3)/96 if p is odd and q is even.
T(p,q) = p^2*q*(2q-3p-6+6pq+6q^2+2qp^2+3pq^2+2q^3)/96 if p is even and q is odd.
The first Maple program makes use of the above formulas.
The Hosoya-Wiener polynomial of C(p) X C(q) is 2*h(p)*h(q) + p*h(q) + q*h(p), where h(j) denotes the Hosoya-Wiener polynomial of the cycle C(j).
The command H(p,q) in the 2nd Maple program yields the Hosoya-Wiener polynomial.
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MAPLE
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HWWi := proc (p, q) if `mod`(p, 2) = 1 and `mod`(q, 2) = 1 then (1/96)*p*q*(3*p^2*q^2+2*p^3*q+2*p*q^3-4*p*q-3*p^2-3*q^2-6*p-6*q+6*p^2*q+6*p*q^2+3) elif `mod`(p, 2) = 0 and `mod`(q, 2) = 0 then (1/96)*p^2*q^2*(6*q+6*p+3*p*q+2*p^2+2*q^2+8) elif `mod`(p, 2) = 1 and `mod`(q, 2) = 0 then (1/96)*p*q^2*(3*p^2*q+2*p^3+2*p*q^2+2*p-3*q-6+6*p^2+6*p*q) else (1/96)*p^2*q*(3*p*q^2+2*q^3+2*p^2*q+2*q-3*p-6+6*q^2+6*p*q) end if end proc: for i to 10 do seq(HWWi(i, j), j = 1 .. i) end do; # yields sequence in triangular form
H := proc (p, q) local br, h: br := proc (n) options operator, arrow: sum(t^k, k = 0 .. n-1) end proc; h := proc (m) if `mod`(m, 2) = 0 then m*(br((1/2)*m)-1)+(1/2)*m*t^((1/2)*m) else m*t*br((1/2)*m-1/2) end if end proc: sort(expand(2*h(p)*h(q)+p*h(q)+q*h(p))) end proc: Wi := proc (p, q) options operator, arrow: subs(t = 1, diff(H(p, q), t)) end proc: for i to 10 do seq(Wi(i, j), j = 1 .. i) end do; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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