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.

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, 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, 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

Offset

0,7

Comments

Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), 0<=k<=n) = F(n+3)-2 = A001911(n).

Links

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Formula

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)].

a(n) = A007298(n+4) - A007298(n+3). - Altug Alkan, May 03 2016

Example

T(6,2) = 3 because we have 110110, 110111, 110101.
Triangle starts:
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;

Maple

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

Mathematica

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 *)

Prog

(Haskell)
a129713 n k = a129713_tabl !! n !! k
a129713_row n = a129713_tabl !! n
a129713_tabl = [1] : [1, 1] : f [1] [1, 1] where
   f us vs = ws : f vs ws where
             ws = zipWith (+) (init us ++ [0, 0, 0]) (vs ++ [1])
-- Reinhard Zumkeller, May 26 2015

Crossrefs

Cf. A000045, A001911.

Cf. A054123.

Cf. A007298. - Altug Alkan, May 03 2016

Sequence in context: A362648 A333893 A225630 * A096669 A096591 A316074

Adjacent sequences: A129710 A129711 A129712 * A129714 A129715 A129716

Keyword

nonn,tabl,easy

Author

Emeric Deutsch, May 12 2007

Status

approved