This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(19) = 1 + 4 + 12 + 36 + 108 + 324 + 948 + 2796 + 8196 + 24060 + 70188 + 205284 + 597996 + 1744548 + 5073900 + 14774652 + 42922452 + 124814484 + 362267652 + 1052271732 = 1604779373 is prime .
a(1) = 4. These are the four directions one can step away from a point on a 2D square lattice.
a(2) = 12. These consist of the two following walks:
.
*
| 1 2
. 2 *---*---.---*
|
*---*
1
.
The first walk can be taken in 8 different ways, the second in 4 ways, giving a total of 12 walks.
a(3) = 36. These consist of the following five walks:
.
* *
| |
. 3 3 .
| 3 *---.---.---* *---.---.---* | 3
. | | .
| . 2 . 2 |
* | | *---*---.---*
| *---* *---* 1 2
. 2 1 1
| *---*---.---*---.---.---*
*---* 1 2 3
1
.
The first four can be taken in 8 different ways, while the last straight walk can be taken in 4 ways, giving a total of 36 walks. Notice it is not possible to form a collision from any of these walks by adding a step of length 4.
Initial rows (paths of length 0, 2, 4, ...):
{ 1 };
{ 4, 4 };
{ 28, 24, 16 };
{ 188, 188, 128, 64 }.
Initial rows (paths of length 1, 3, 5, ...):
{ 2 };
{ 10, 8 };
{ 74, 56, 32 };
{ 518, 464, 288, 128 }.
a(1) = 4. These are the four directions one can step 1 unit away from the origin on a 2D square lattice.
a(2) = 24. These consist of the following four walks:
.
*
| * 1 2 2 1
. 2 | 1 *---*---.---* *---.---*---*
| *---.---*
*---* 2
1
.
The first two can be walked in eight different ways on a 2D lattice, the last two in four different ways, giving a total of 2*8+2*4 = 24.
a(3) = 216. Restricting the first step to the right then the different ways a walk can take three steps on a 2D lattice within the first quadrant are RUL, RUU, RUR, RRU, RRR. Each of these can be taken in 6 ways, the arrangements of 1,2,3. The first four walks can also be taken in eight ways on the 2D lattice, the last in four ways, giving a total of 4*8*3!+1*4*3! = 216.
a(4) = 2544. Restricting the first step to the right then the different ways a walk can take four steps on a 2D lattice within the first quadrant are RULD, RULL, RULU, RUUL, RUUU, RUUR, RURU, RURR, RURD, RRUL, RRUU, RRUR, RRRU, RRRR. Each of these can be taken in 24 ways, the arrangements of 1,2,3,4. However six of these walks are forbidden due to the collisions given in the comments. The first thirteen walks can also be taken in eight ways on the 2D lattice, the fourteenth in four ways. This gives a total number of walks of 13*8*4! - 6*8 + 4*4! = 2544.
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