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A245964 Triangle read by rows: T(n,k) is the number of maximal hypercubes Q(p) in the Lucas cube Lambda(n) (i.e., Q(p) is an induced subgraph of Lambda(n) that is not a subgraph of a subgraph of Lambda(n) that is isomorphic to the hypercube Q(p+1)).

%I A245964 #21 Jul 21 2019 06:50:10
%S A245964 1,1,0,2,0,3,0,0,2,0,0,5,0,0,3,2,0,0,0,7,0,0,0,8,2,0,0,0,3,9,0,0,0,0,
%T A245964 15,2,0,0,0,0,11,11,0,0,0,0,3,24,2,0,0,0,0,0,26,13,0,0,0,0,0,14,35,2,
%U A245964 0,0,0,0,0,3,50,15,0,0,0,0,0,0,40,48,2,0,0,0,0,0,0,17,85,17
%N A245964 Triangle read by rows: T(n,k) is the number of maximal hypercubes Q(p) in the Lucas cube Lambda(n) (i.e., Q(p) is an induced subgraph of Lambda(n) that is not a subgraph of a subgraph of Lambda(n) that is isomorphic to the hypercube Q(p+1)).
%C A245964 Row n contains 1 + floor(n/2) entries.
%C A245964 Sum of entries in row n = A001608(n) (n>=2) (the Perrin sequence).
%H A245964 S. Klavzar, <a href="http://dx.doi.org/10.1007/s10878-011-9433-z">Structure of Fibonacci cubes: a survey</a>, J. Comb. Optim., 25, 2013, 505-522.
%H A245964 M. Mollard, <a href="http://arxiv.org/abs/1201.1494">Maximal hypercubes in Fibonacci and Lucas cubes</a>, arXiv:1201.1494 [math.CO], 2012.
%H A245964 M. Mollard, <a href="http://dx.doi.org/10.1016/j.dam.2012.06.003">Maximal hypercubes in Fibonacci and Lucas cubes</a>, Discrete Appl. Math., 160, 2012, 2479-2483.
%F A245964 T(n,k) = (n/k)*binomial(k,n-2k) (k>=1).
%F A245964 G.f.: (1+z+t*z^2+t*z^3-t*z^4)/(1-t*(1+z)*z^2).
%e A245964 Row 4 is 0,0,2. Indeed, the Lucas cube Lambda(4) is the graph <><> obtained by identifying a vertex of a square with a vertex of another square; each square is a maximal Q(2).
%e A245964 Triangle starts:
%e A245964   1;
%e A245964   1;
%e A245964   0, 2;
%e A245964   0, 3;
%e A245964   0, 0, 2;
%e A245964   0, 0, 5;
%e A245964   0, 0, 3, 2;
%e A245964   0, 0, 0, 7;
%e A245964   0, 0, 0, 8, 2;
%p A245964 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 else n*binomial(k, n-2*k)/k end if end proc: for n from 0 to 20 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%t A245964 T[0|1, 0] = 1; T[_, 0] = 0; T[n_, k_] := (n/k)*Binomial[k, n - 2k];
%t A245964 Table[T[n, k], {n, 0, 20}, {k, 0, n/2}] // Flatten (* _Jean-François Alcover_, Dec 01 2017 *)
%Y A245964 Cf. A001608, A245963.
%K A245964 nonn,tabf
%O A245964 0,4
%A A245964 _Emeric Deutsch_, Aug 13 2014