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A180867
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Square array read by antidiagonals: T(m,n) is the Wiener index of the Dutch windmill graph D(m,n) (m>=3, n>=1).
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2
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3, 8, 14, 15, 40, 33, 27, 78, 96, 60, 42, 144, 189, 176, 95, 64, 228, 351, 348, 280, 138, 90, 352, 558, 648, 555, 408, 189, 125, 500, 864, 1032, 1035, 810, 560, 248, 165, 700, 1230, 1600, 1650, 1512, 1113, 736, 315, 216, 930, 1725, 2280, 2560, 2412, 2079, 1464
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OFFSET
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3,1
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COMMENTS
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The Dutch windmill graph D(m,n) (also called friendship graph) is the graph obtained by taking n copies of the cycle graph C_m with a vertex in common (i.e., a bouquet of n C_m graphs).
The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
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LINKS
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FORMULA
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T(m,n) = (1/8)n(m^2-1)(2mn-m-2n+2) if m is odd.
T(m,n) = (1/8)n*m^2*(2mn-m-2n+2) if m is even.
The Wiener polynomial of D(m,n) is n*w(m)+(1/2)n(n-1)r(m)^2, where, denoting S(t,p) = Sum_{j=1..p-1}t^j, we have w(m) = mS(t,m/2) + (1/2)mt^(m/2), r(m) = 2S(t,m/2) + t^(m/2) if m is even and w(m) = mS(t,(m+1)/2), r(m) = 2S(t,(m+1)/2) if m is odd.
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EXAMPLE
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Square array starts:
3, 14, 33, 60, 95, ...
8, 40, 96, 176, 280, ...
15, 78, 189, 348, 555, ...
27, 144, 351, 648, 1035, ...
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MAPLE
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T := proc (m, n) if `mod`(m, 2) = 1 then (1/8)*n*(m^2-1)*(2*m*n-m-2*n+2) else (1/8)*n*m^2*(2*m*n-m-2*n+2) end if end proc: for n from 3 to 12 do seq(T(n+1-j, j), j = 1 .. n-2) 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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EXTENSIONS
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Typos in Wiener polynomial information corrected by Emeric Deutsch, Sep 30 2010
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STATUS
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approved
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