A078805 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's, every runlength of 1's odd and initial letter 0.

1, 1, 1, 1, 2, 0, 1, 3, 1, 1, 1, 4, 3, 2, 0, 1, 5, 6, 4, 2, 1, 1, 6, 10, 8, 6, 2, 0, 1, 7, 15, 15, 13, 6, 3, 1, 1, 8, 21, 26, 25, 16, 9, 2, 0, 1, 9, 28, 42, 45, 36, 22, 9, 4, 1, 1, 10, 36, 64, 77, 72, 50, 28, 12, 2, 0, 1, 11, 45, 93, 126, 133, 106, 70, 34, 13, 5, 1, 1, 12, 55, 130, 198, 232

Offset

1,5

References

Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151.

Links

Table of n, a(n) for n=1..84.

Formula

T(n, k)=T(n-2, k)+T(n-2, k-1)+T(n-2, k-2)+T(n-3, k-1)-T(n-4, k-2) for 0<=k<=n, n>=4. (All numbers T(i, j) not in the array are 0, by definition of T.)

Example

T(5,2) counts the words 01010, 01001, 00101.
Top of triangle T:
  1 = T(1,0)
  1 1 = T(2,0) T(2,1)
  1 2 0 = T(3,0) T(3,1) T(3,2)
  1 3 1 1
  1 4 3 2 0

Mathematica

Clear[t];
t[n_, k_]:=0/; Not[0<=n<=200&&0<=k<=200];
t[1, 0]=1; t[2, 0]=1; t[2, 1]=1;
t[3, 0]=1; t[3, 1]=2; t[3, 2]=0; t[4, 2]=1;
t[n_, k_]:=t[n, k]=t[n-2, k]+t[n-2, k-1]+t[n-2, k-2]+t[n-3, k-1]-t[n-4, k-2]
u=Table[t[n, k], {n, 1, 12}, {k, 0, n-1}]
Grid[u] (* triangle *)
Flatten[u] (* sequence *)
(* Clark Kimberling Jul 18 2025 *)

Crossrefs

Cf. A078804, A078806.

Row sums: A028495.

Sequence in context: A093555 A065432 A094184 * A122837 A143359 A291316

Adjacent sequences: A078802 A078803 A078804 * A078806 A078807 A078808

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 07 2002

Status

approved