login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A171437 Square array read by antidiagonals: T(m,n) is the Wiener index of the prism graph C(m) x P(n) (m>=3, n>=1). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph. 0
3, 8, 21, 15, 48, 63, 27, 85, 136, 138, 42, 144, 235, 288, 255, 64, 217, 387, 490, 520, 423, 90, 320, 574, 792, 875, 848, 651, 125, 441, 832, 1162, 1395, 1415, 1288, 948, 165, 600, 1134, 1664, 2030, 2232, 2135, 1856, 1323, 216, 781, 1525, 2250, 2880, 3227 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

T(m,2)=A138179(m).

REFERENCES

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

LINKS

Table of n, a(n) for n=3..53.

Eric Weisstein's World of Mathematics, Prism Graph.

FORMULA

T(m,n)=(1/24)mn([4m(n^2-1)+3n(m^2-1)] if m is odd.

T(m,n)=(1/24)m^2*n[4(n^2-1)+3mn] if m is even.

The Wiener polynomial p[n](t) for C(2m+1) x P(n) satisfies the recurrence relation p[n]=p[n-1]+p[1]+(2m+1)*a[n], where a[n]=[t+2*sum(t^j,j=2..m+1)*sum(t^j,j=0..n-2) and p[1]=(2m+1)*sum(t^j,j=1..m).

The Wiener polynomial q[n](t) for C(2m) x P(n) satisfies the recurrence relation q[n]=q[n-1]+q[1]+2m*b[n], where b[n]=[t+t^{m+1}+2*sum(t^j,j=2..m)*sum(t^j,j=0..n-2) and q[1]=2m*sum(t^j,j=1..m-1)+m*t^m.

The above recurrence relations yield simple expressions for the Wiener polynomials (however, they are too long for being reproduced here).

EXAMPLE

Square array starts:

3,21,63,138,255,...

8,48,136,288,520,...

15,85,235,490,875,...

27,144,387,792,1395,...

MAPLE

T := proc (m, n) if `mod`(m, 2) = 1 then (1/24)*m*n*(4*m*(n^2-1)+3*n*(m^2-1)) else (1/24)*m^2*n*(4*n^2-4+3*m*n) end if end proc: for m from 3 to 12 do seq(T(m+1-j, j), j = 1 .. m-2) end do; # yields sequence in triangular form

CROSSREFS

Cf. A138179.

Sequence in context: A027320 A027319 A066212 * A075719 A245205 A101643

Adjacent sequences:  A171434 A171435 A171436 * A171438 A171439 A171440

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Sep 29 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 07:21 EDT 2017. Contains 284250 sequences.