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A180860 Square array read by antidiagonals: T(m,n) is the Wiener index of the tadpole graph L(m,n) (m>=3, n>=1). L(m,n) is the graph obtained by joining with an edge a node in the cycle graph C_m to an end-node of the path graph P_n. The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph. 1
8, 16, 17, 26, 29, 31, 42, 43, 48, 51, 61, 64, 67, 74, 78, 88, 88, 94, 99, 108, 113, 119, 121, 124, 133, 140, 151, 157, 160, 158, 164, 170, 182, 191, 204, 211, 206, 206, 208, 218, 227, 242, 253, 268, 276, 264, 259, 264, 270, 284, 296, 314, 327, 344, 353, 328 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

Tadpole graphs are also called keys (see the Gross & Yellen reference, p. 895).

T(m,1)=A180861(m).

REFERENCES

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004.

LINKS

Table of n, a(n) for n=3..58.

Eric Weisstein's World of Mathematics, Tadpole Graph.

FORMULA

T(2m,n)=m^3+(1/6)n(n^2-1)+mn(m+n+1).

T(2m+1,n)=(1/2)m(m+1)(2m+1)+(1/6)n(n+1)(n+2)+mn(m+n+2).

EXAMPLE

T(3,1)=8 because the graph consists of a triangle ABCA and an edge AD; the distances are d(A,B)=d(B,C)=d(C,A)=d(A,D)=1 and d(DB)=d(DC)=2.

Square array T(i,j) begins:

8,17,31,51,78,...

16,29,48,74,108,...

26,43,67,99,140,...

42,64,94,133,182,...

MAPLE

T := proc (m, n) if `mod`(m, 2) = 0 then (1/8)*m^3+(1/6)*n*(n^2-1)+(1/4)*m*n*(m+2*n+2) else (1/8)*m*(m^2-1)+(1/6)*n*(n+1)*(n+2)+(1/4)*n*(m-1)*(m+2*n+3) end if end proc: for n from 3 to 13 do seq(T(n+1-i, i), i = 1 .. n-2) end do; # yields sequence in triangular form

CROSSREFS

Cf. A180861

Sequence in context: A297141 A004779 A247061 * A328189 A101766 A338930

Adjacent sequences:  A180857 A180858 A180859 * A180861 A180862 A180863

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Sep 27 2010

STATUS

approved

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Last modified May 25 00:03 EDT 2022. Contains 354047 sequences. (Running on oeis4.)