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A026768 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026758.

%I A026768 #9 Nov 01 2019 03:54:42
%S A026768 1,1,2,3,6,9,16,29,46,82,145,237,421,737,1228,2171,3788,6388,11253,
%T A026768 19617,33344,58597,102141,174571,306294,533976,916309,1605975,2800260,
%U A026768 4820020,8441365,14721208,25399974,44458045,77542951
%N A026768 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026758.
%H A026768 G. C. Greubel, <a href="/A026768/b026768.txt">Table of n, a(n) for n = 0..1000</a>
%p A026768 T:= proc(n,k) option remember;
%p A026768    if n<0 then 0;
%p A026768    elif k=0 or k = n then 1;
%p A026768    elif type(n,'odd') and k <= (n-1)/2 then
%p A026768         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026768    else
%p A026768        procname(n-1,k-1)+procname(n-1,k) ;
%p A026768    end if ;
%p A026768 end proc;
%p A026768 seq( add(T(n-k,k), k=0..floor(n/2)), n=0..30); # _G. C. Greubel_, Oct 31 2019
%t A026768 T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k<=(n - 1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]; Table[Sum[T[n-k,k], {k,0,Floor[n/2]}], {n, 0, 30}] (* _G. C. Greubel_, Oct 31 2019 *)
%o A026768 (Sage)
%o A026768 @CachedFunction
%o A026768 def T(n, k):
%o A026768     if (n<0): return 0
%o A026768     elif (k==0 or k==n): return 1
%o A026768     elif (mod(n,2)==1 and k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
%o A026768     else: return T(n-1,k-1) + T(n-1,k)
%o A026768 [sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, Oct 31 2019
%Y A026768 Cf. A026758, A026759, A026760, A026761, A026762, A026763, A026764, A026765, A026766, A026767.
%K A026768 nonn
%O A026768 0,3
%A A026768 _Clark Kimberling_