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(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.
For n = 2, total steps = 4, there are 8 different paths with an area of 1. These are the 8 possible ways to walk the polygon:
+---+
| |
+---+
.
For n = 3, total steps = 6, there are 24 different paths with an area of 2. These are the 24 possible ways to walk the polygon:
+---+---+
| |
+---+---+
.
For n = 4, total steps = 8, there are 96 different paths with an area of 3 and 16 different paths with an area of 4. These are the possible ways to walk the polygons:
+---+ +---+---+
| | | |
+ +---+ + +
| | | |
+---+---+ for area = 3 +---+---+ for area = 4
.
For n = 5, total steps = 10, there are 360 different paths with an area of 4, 160 paths with an area of 5 and 40 different paths with an area of 6. These are the possible ways to walk the polygons:
+---+---+---+---+ +---+ +---+ +---+---+
| | | | | | | |
+---+---+---+---+ + +---+---+ +---+ +---+ +---+ +---+
| | | | | |
+---+---+---+ +---+---+---+ +---+---+ for area = 4
.
+---+---+ +---+---+---+
| | | |
+ +---+ + +
| | | |
+---+---+---+ for area = 5 +---+---+---+ for area = 6
.
Table begins:
0;
8;
24;
96,16;
360,160,40;
1320,960,528,144,24;
4872,4704,3752,2016,840,224,56;
18112,21632,20992,15424,9920,4832,2176,704,192,32;
67248,96192,107712,93312,75096,50112,31104,16416,7848,3168,1080,288,72;
249480,415040,526400,514480,468680,373280,281280,189920,120400,69120,36560,17040,7480,2720,880,240,40;
Row sums = A010566.
a(1) = 4. These are the four ways one can step away from the origin on a 2D square lattice.
a(2) = 12. These consist of the two following walks:
.
*
|
.
| 3 2 3
. *---.---*---.---.---*
|
*---.---*
2
.
The first walk can be taken in eight different ways on the 2D square lattice, the second in four ways, giving a total of 12 walks.
a(7) = 2876. If we consider only walks starting with one or more steps to the right followed by an upward step, and ignoring collisions, then the total number of walks is 3^5+3^4+3^3+3^2+3^1+3^0 = 364. However, five of these are forbidden due to the collisions given in the comments, leaving 359 in total. These can be walked in eight different ways on the 2D grid. There are also the four straight walks along the axes. This gives a total of 359*8+4 = 2876 walks.
a(0..3) are the same as the standard square lattice SAW of A001411 as the walk cannot step to a smaller ring in the first three steps. a(4) = 92. If we restrict the first one or more steps to the right followed by an upward step then there is one walk which steps to a smaller ring and is thus forbidden. That is the walk (0,0) -> (1,0) -> (2,0) -> (2,1) -> (1,1). As this can be walked in eight different ways on the square lattice the number of 4-step walks becomes A001411(4) - 8 = 100 - 8 = 92.
A076874:=n->n - floor((4*n + 1)^(1/2)); seq(A076874(n), n=3..50); # Wesley Ivan Hurt, Mar 01 2014
Table[n - Floor[(4 n + 1)^(1/2)], {n, 3, 50}] (* Wesley Ivan Hurt, Mar 01 2014 *)
a(1) = 4 as a single step of length 1 can be taken in four ways on the square lattice the sum of square end-to-end displacements is 4*1 = 4.
a(2) = 32. The two 2-step self-avoiding walks with a first step to the right in the first quadrant with their corresponding square displacements are:
.
+
| 2 +---+---+ 4
+---+
.
The first walk can be taken in 8 ways on a square lattice, the latter in 4 ways, thus the total displacement over all 2-step walks is 8*2 + 4*4 = 32.
a(3) = 164. The five 3-step self-avoiding walks with a first step to the right in the first quadrant with their corresponding square displacements are:
.
+
+---+ | +---+ +
| 1 + 5 | 5 | 5 +---+---+---+ 9
+---+ | +---+ +---+---+
+---+
.
The first four walks can be taken in 8 ways on a square lattice, the last in 4 ways, thus the total displacement over all 3-step walks is 8*1 + 8*5 + 8*5 + 8*5 + 4*9 = 164.
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:
.
*
|
.
|
. 4
| 1 4
. *---*---.---.---.---*
|
*---*
1
.
The first walk can be taken in 8 different ways, the second in 4 ways, giving a total of 12 walks.
a(8) = 8676. If we consider only walks starting with one or more steps to the right followed by an upward step, and ignoring collisions, then the total number of walks is 3^6 + 3^5 + 3^4 + 3^3 + 3^2 + 3^1 + 3^0 = 1093. However, nine of these are forbidden due to the collisions given in the comments, leaving 1084 in total. These can be walked in eight different ways on the 2D grid. There are also the four straight walks along the axes. This gives a total of 1084*8 + 4 = 8676 walks.
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