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A129713 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and starting with exactly k 1's (0<=k<=n). A Fibonacci binary word is a binary word having no 00 subword.

%I A129713 #14 Sep 16 2017 17:20:02
%S A129713 1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,5,3,2,1,1,1,8,5,3,2,1,1,1,13,8,5,3,2,1,
%T A129713 1,1,21,13,8,5,3,2,1,1,1,34,21,13,8,5,3,2,1,1,1,55,34,21,13,8,5,3,2,1,
%U A129713 1,1,89,55,34,21,13,8,5,3,2,1,1,1,144,89,55,34,21,13,8,5,3,2,1,1,1,233,144
%N A129713 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and starting with exactly k 1's (0<=k<=n). A Fibonacci binary word is a binary word having no 00 subword.
%C A129713 Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), 0<=k<=n) = F(n+3)-2 = A001911(n).
%H A129713 Reinhard Zumkeller, <a href="/A129713/b129713.txt">Rows n = 0..125 of triangle, flattened</a>
%F A129713 T(n,k) = F(n-k) if k<=n-2, T(n,n-1) = T(n,n) = 1, where F(j) are the Fibonacci numbers (F(0)=0, F(1)=1). G.f.: G(t,z) = (1-z^2)/[(1-z-z^2)(1-tz)].
%F A129713 a(n) = A007298(n+4) - A007298(n+3). - _Altug Alkan_, May 03 2016
%e A129713 T(6,2) = 3 because we have 110110, 110111, 110101.
%e A129713 Triangle starts:
%e A129713 1;
%e A129713 1,1;
%e A129713 1,1,1;
%e A129713 2,1,1,1;
%e A129713 3,2,1,1,1;
%e A129713 5,3,2,1,1,1;
%e A129713 8,5,3,2,1,1,1;
%p A129713 with(combinat): T:=proc(n,k) if k<=n-2 then fibonacci(n-k) elif k=n-1 or k=n then 1 else 0 fi end: for n from 0 to 15 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A129713 nn=15;a=1/(1-y x);b=1/(1-x);Map[Select[#,#>0&]&,CoefficientList[Series[a (1+x)/(1-x^2b),{x,0,nn}],{x,y}]]//Grid (* _Geoffrey Critzer_, Dec 04 2013 *)
%o A129713 (Haskell)
%o A129713 a129713 n k = a129713_tabl !! n !! k
%o A129713 a129713_row n = a129713_tabl !! n
%o A129713 a129713_tabl = [1] : [1, 1] : f [1] [1, 1] where
%o A129713    f us vs = ws : f vs ws where
%o A129713              ws = zipWith (+) (init us ++ [0, 0, 0]) (vs ++ [1])
%o A129713 -- _Reinhard Zumkeller_, May 26 2015
%Y A129713 Cf. A000045, A001911.
%Y A129713 Cf. A054123.
%Y A129713 Cf. A007298. - _Altug Alkan_, May 03 2016
%K A129713 nonn,tabl,easy
%O A129713 0,7
%A A129713 _Emeric Deutsch_, May 12 2007