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A228313
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Triangle read by rows: T(p,q) (1<=q<=p) is the Wiener index of the Cartesian product of the cycles C(p) and C(q) (the torus grid graph).
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1
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0, 1, 8, 3, 21, 54, 8, 48, 120, 256, 15, 85, 210, 440, 750, 27, 144, 351, 720, 1215, 1944, 42, 217, 525, 1064, 1785, 2835, 4116, 64, 320, 768, 1536, 2560, 4032, 5824, 8192, 90, 441, 1053, 2088, 3465, 5427, 7812, 10944, 14580, 125, 600, 1425, 2800
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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(p+q)(pq - 1)/8 if both p and q are odd.
T(p,q) = p^2*q^2*(p + q)/8 if both p and q are even.
T(p,q) = pq^2*(p^2 - 1 + pq)/8 if p is odd and q is even.
T(p,q) = p^2*q*(q^2 - 1 + pq)/8 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 corresponding Hosoya-Wiener polynomial.
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MAPLE
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Wi := proc (p, q) if `mod`(p, 2) = 1 and `mod`(q, 2) = 1 then (1/8)*p*q*(p+q)*(p*q-1) elif `mod`(p, 2) = 0 and `mod`(q, 2) = 0 then (1/8)*p^2*q^2*(p+q) elif `mod`(p, 2) = 1 and `mod`(q, 2) = 0 then (1/8)*p*q^2*(p^2+p*q-1) else (1/8)*p^2*q*(q^2+p*q-1) end if end proc: for i to 10 do seq(Wi(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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