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%I #25 Feb 26 2017 04:34:25
%S 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,
%T 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,
%U 10,2,0,0,0,0,0,1,11,55,77,44,11,0
%N Triangle read by rows: T(n,k) = number of vertices of eccentricity k in the Lucas cube graph of order n.
%C 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
%H Alois P. Heinz, <a href="/A210572/b210572.txt">Rows n = 0..140, flattened</a>
%H A. Castro and M. Mollard, <a href="http://dx.doi.org/10.1016/j.disc.2011.11.006">The eccentricity sequences of Fibonacci and Lucas cubes</a>, Discrete Math., 312 (2012), 1025-1037.
%F 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
%e 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.
%e Triangle begins:
%e 1,
%e 1, 0,
%e 0, 1, 2,
%e 0, 1, 3, 0,
%e 0, 0, 1, 4, 2,
%e 0, 0, 1, 5, 5, 0,
%e 0, 0, 0, 1, 9, 6, 2,
%e 0, 0, 0, 1, 7, 14, 7, 0,
%e 0, 0, 0, 0, 1, 16, 20, 8, 2,
%e 0, 0, 0, 0, 1, 9, 30, 27, 9, 0,
%e ...
%p 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;
%p # yields sequence in triangular form - _Emeric Deutsch_, Aug 06 2014
%t 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_ *)
%K nonn,tabl
%O 0,6
%A _N. J. A. Sloane_, Mar 22 2012
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