|
| |
|
|
A210572
|
|
Triangle read by rows: T(n,k) = number of vertices of eccentricity k in the Lucas cube graph of order n.
|
|
2
|
|
|
|
1, 1, 0, 0, 1, 2, 0, 1, 3, 0, 0, 0, 1, 4, 2, 0, 0, 1, 5, 5, 0, 0, 0, 0, 1, 9, 6, 2, 0, 0, 0, 1, 7, 14, 7, 0, 0, 0, 0, 0, 1, 16, 20, 8, 2, 0, 0, 0, 0, 1, 9, 30, 27, 9, 0, 0, 0, 0, 0, 0, 1, 25, 50, 35, 10, 2, 0, 0, 0, 0, 0, 1, 11, 55, 77, 44, 11, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,6
|
|
|
COMMENTS
|
The Castro and Mollard reference contains a formula for T(n,k) (Corollary 5.17); it is used in the Maple program given below. - Emeric Deutsch, Aug 06 2014
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: (1 + t*z^2)/(1 - t*z - t*z^2) + 1/(1 + t*z) - (1 - z)/(1 - t*z^2). - Emeric Deutsch, Aug 06 2014
|
|
|
EXAMPLE
|
Row 3 is 0,1,3,0 because the Lucas cube L_3 is the star tree on 4 vertices, having eccentricities 1, 2, 2, 2.
Triangle begins:
1,
1, 0,
0, 1, 2,
0, 1, 3, 0,
0, 0, 1, 4, 2,
0, 0, 1, 5, 5, 0,
0, 0, 0, 1, 9, 6, 2,
0, 0, 0, 1, 7, 14, 7, 0,
0, 0, 0, 0, 1, 16, 20, 8, 2,
0, 0, 0, 0, 1, 9, 30, 27, 9, 0,
...
|
|
|
MAPLE
|
T := proc (n, k) if n = 0 and k = 0 then 1 elif n = 1 and k = 0 then 1 elif k = 0 then 0 elif `mod`(n, 2) = 0 and k = n then 2 elif `mod`(n, 2) = 1 and k = n then 0 elif n = 2*k then binomial(k, n-k)+binomial(k-1, n-k-1)-1 elif n = 2*k+1 then binomial(k, n-k)+binomial(k-1, n-k-1)+1 else binomial(k, n-k)+binomial(k-1, n-k-1) end if end proc: for m from 0 to 16 do seq(T(m, k), k = 0 .. m) end do;
|
|
|
MATHEMATICA
|
T[n_, k_] := Which[n == 0 && k == 0, 1, n == 1 && k == 0, 1, k == 0, 0, EvenQ[n] && k == n, 2, OddQ[n] && k == n, 0, n == 2*k, Binomial[k, n-k] + Binomial[k-1, n-k-1]-1, n == 2*k+1, Binomial[k, n-k] + Binomial[k-1, n-k- 1]+1, True, Binomial[k, n-k] + Binomial[k-1, n-k-1]]; Table[T[m, k], {m, 0, 16}, {k, 0, m}] // Flatten (* Jean-François Alcover, Feb 26 2017, after Emeric Deutsch *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
