login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A284652 Number T(n,k) of self-avoiding planar walks of length k starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal; triangle T(n,k), k>=0, floor((sqrt(1+8*k)-1)/2)<=n<=k, read by columns. 3
1, 1, 1, 2, 1, 4, 1, 4, 9, 1, 4, 8, 21, 7, 16, 22, 51, 3, 21, 54, 54, 127, 1, 17, 87, 178, 142, 323, 1, 15, 87, 269, 565, 370, 835, 10, 116, 370, 896, 1766, 983, 2188, 9, 99, 499, 1473, 2776, 5446, 2627, 5798, 4, 91, 536, 2290, 5528, 8657, 16655, 7086, 15511 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Columns k = 0..160, flattened

Wikipedia, Lattice path

Wikipedia, Self-avoiding walk

FORMULA

Sum_{k=n..n*(n+3)/2} (k+1) * T(n,k) = A284231(n).

EXAMPLE

Triangle T(n,k) begins:

1;

.  1, 1;

.  .  2, 1, 1,  1;

.  .  .  4, 4,  4,  7,   3,   1,   1;

.  .  .  .  9,  8, 16,  21,  17,  15,   10,    9, ... ;

.  .  .  .  .  21, 22,  54,  87,  87,  116,   99, ... ;

.  .  .  .  .   .  51,  54, 178, 269,  370,  499, ... ;

.  .  .  .  .   .   .  127, 142, 565,  896, 1473, ... ;

.  .  .  .  .   .   .    .  323, 370, 1766, 2776, ... ;

.  .  .  .  .   .   .    .    .  835,  983, 5446, ... ;

.  .  .  .  .   .   .    .    .     . 2188, 2627, ... ;

CROSSREFS

Row sums give A284230.

Column sums give A284415.

Antidiagonal sums give A284428.

T(n,n) gives A001006.

T(n,n+1) gives A284778.

T(n,2n) gives A284416.

T(n,n*(n+1)/2) gives A284418.

Column heights give A122797(k+1).

Cf. A000096, A284231, A284461, A284414 (this triangle read by rows).

Sequence in context: A007104 A102627 A296560 * A261242 A088296 A282738

Adjacent sequences:  A284649 A284650 A284651 * A284653 A284654 A284655

KEYWORD

nonn,tabf,walk

AUTHOR

Alois P. Heinz, Mar 31 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 25 18:35 EST 2020. Contains 332256 sequences. (Running on oeis4.)