A274228 Triangle read by rows: T(n,k) (n>=3, 0<=k<=n-3) = number of n-sequences of 0's and 1's with exactly one pair of adjacent 0's and exactly k pairs of adjacent 1's.

2, 3, 2, 4, 4, 2, 5, 8, 5, 2, 6, 12, 12, 6, 2, 7, 18, 21, 16, 7, 2, 8, 24, 36, 32, 20, 8, 2, 9, 32, 54, 60, 45, 24, 9, 2, 10, 40, 80, 100, 90, 60, 28, 10, 2, 11, 50, 110, 160, 165, 126, 77, 32, 11, 2, 12, 60, 150, 240, 280, 252, 168, 96, 36, 12, 2, 13, 72, 195, 350, 455, 448, 364, 216, 117, 40, 13, 2

Offset

3,1

Links

Table of n, a(n) for n=3..80.

Formula

T(n,k) = (k+1)*(binomial(floor((n+k-2)/2),k+1)+binomial(floor((n+k-3)/2),k+1))+2*binomial(floor((n+k-3)/2),k).

T(n,k) = (k+1)*A073044(n-2,k+1) + 2*A046854(n-3,k).

T(n,k) = A274742(n,k)+A274742(n-1,k)+A046854(n-3,k).

Example

n=3 => 100, 001 -> T(3,0) = 2.
n=4 => 0010, 0100, 1001 -> T(4,0) = 3; 0011, 1100 -> T(4,1) = 2.
Triangle starts:
2,
3, 2,
4, 4, 2,
5, 8, 5, 2,
6, 12, 12, 6, 2,
7, 18, 21, 16, 7, 2,
8, 24, 36, 32, 20, 8, 2,
9, 32, 54, 60, 45, 24, 9, 2,
10, 40, 80, 100, 90, 60, 28, 10, 2,
11, 50, 110, 160, 165, 126, 77, 32, 11, 2,
12, 60, 150, 240, 280, 252, 168, 96, 36, 12, 2,
13, 72, 195, 350, 455, 448, 364, 216, 117, 40, 13, 2,
...

Mathematica

Table[(k + 1) (Binomial[Floor[(n + k - 2)/2], k + 1] + Binomial[Floor[(n + k - 3)/2], k + 1]) + 2 Binomial[Floor[(n + k - 3)/2], k], {n, 3, 14}, {k, 0, n - 3}] // Flatten (* Michael De Vlieger, Jun 16 2016 *)

Prog

(PARI) T(n, k) = (k+1)*(binomial((n+k-2)\2, k+1)+binomial((n+k-3)\2, k+1))+2*binomial((n+k-3)\2, k); \\ Michel Marcus, Jun 17 2016

Crossrefs

Row sums give A001629.

Cf. A073044.

Columns of table:

T(n,0)=A000027(n-1)

T(n,1)=A007590(n-1)

T(n,2)=A080838(n-1)

T(n,3)=A032091(n)

Sequence in context: A338162 A215182 A214906 * A321476 A214567 A259363

Adjacent sequences: A274225 A274226 A274227 * A274229 A274230 A274231

Keyword

nonn,tabl

Author

Jeremy Dover, Jun 14 2016

Status

approved