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Search: id:A143940
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| A143940 |
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Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in a linear chain of n triangles (i.e. joined like VVV..VV; here V is a triangle!; 1<=k<=n). |
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+0
2
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| 3, 6, 4, 9, 8, 4, 12, 12, 8, 4, 15, 16, 12, 8, 4, 18, 20, 16, 12, 8, 4, 21, 24, 20, 16, 12, 8, 4, 24, 28, 24, 20, 16, 12, 8, 4, 27, 32, 28, 24, 20, 16, 12, 8, 4, 30, 36, 32, 28, 24, 20, 16, 12, 8, 4, 33, 40, 36, 32, 28, 24, 20, 16, 12, 8, 4, 36, 44, 40, 36, 32, 28, 24, 20, 16, 12, 8, 4
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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The entries in row n are the coefficients of the Wiener polynomial of a linear chain of n triangles.
Sum of entries in row n = n(2n+1)=A014105(n).
Sum(k*T(n,k), k=1..n)=the Wiener index of the linear chain of n triangles = A143941(n).
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REFERENCES
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B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60 (1996), 959-969.
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FORMULA
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T(n,1)=3n; T(n,k)=4(n-k+1) for k>1.
G.f.=G(q,z)=qz/(3+qz)/[(1-qz)*(1-z)^2].
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EXAMPLE
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T(2,1)=6 because the chain of 2 triangles has 6 edges.
Triangle starts:
3;
6,4;
9,8,4;
12,12,8,4;
15,16,12,8,4;
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MAPLE
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T:=proc(n, k) if n < k then 0 elif k = 1 then 3*n else 4*n-4*k+4 end if end proc: for n to 12 do seq(T(n, k), k=1..n) end do; # yields sequence in triangular form
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CROSSREFS
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A014105, A143941
Sequence in context: A100000 A083682 A021278 this_sequence A083349 A065230 A163294
Adjacent sequences: A143937 A143938 A143939 this_sequence A143941 A143942 A143943
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 06 2008
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