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
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
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Sep 27 2010
STATUS
approved