A336262
Number of self-avoiding walks on a 2-dimensional square lattice where the walk consists of steps with incrementing lengths equal to the prime numbers, from 2 to prime(n).
Original entry on oeis.org
1, 4, 12, 36, 108, 324, 972, 2876, 8364, 24124, 69116, 196916, 559604, 1585764, 4495740, 12714796, 35654620, 99686708, 278880060, 781504972, 2180418716, 6079373324, 16857930068, 46773551052, 129562831140, 358157148332
Offset: 0
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:
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*
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| 3 2 3
. *---.---*---.---.---*
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*---.---*
2
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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.
A078527
Number of maximally 2-constrained walks on square lattice trapped after n steps.
Original entry on oeis.org
0, 1, 9, 7, 3, 36, 26, 13, 1, 100, 54, 19, 7, 247, 147, 68, 27, 12, 552, 294, 151
Offset: 7
a(7)=0 because the unique shortest possible self-trapping walk has no constrained steps. Of the A077482(10)=25 self-trapping walks of length n=10, there are A078528(10)=5 unconstrained walks (9 steps with free choice of direction). a(10)=7 walks are maximally 2-constrained containing 2 steps with k=2. Among the remaining 13 walks there are 11 walks having 1 step with k=2 and 2 walks have 1 forced step k=1. An illustration of all unconstrained and all maximally 2-constrained 10-step walks is given in the first link under "5 Unconstrained and 7 maximally 2-constrained walks of length 10". a(15)=1 is a unique ("perfectly constrained") walk visiting all lattice points of a 4*4 square, see "Examples for walks with the maximum number of constrained steps" provided at the given link.
A078799
Sum of square displacements over all self-avoiding walks on square lattice trapped after n steps.
Original entry on oeis.org
1, 6, 35, 150, 627, 2318, 8400, 28624, 96049, 311002, 994899, 3111570, 9638347, 29398762, 88985840, 266359752, 792360385, 2337329116, 6859721431, 20000471236, 58067533570, 167703151726
Offset: 7
a(9)=35 because the A077482(9)=11 different self-trapping walks stop at 5*(0,1)->d^2=1, 2*(1,2)->d^2=5, 2*(2,1)->d^2=5, (-1,0)->d^2=1 (3,0)->d^2=9. a(9)=5*1+2*5+2*5+1+9=35 See "Enumeration of all short self-trapping walks" at link
A078800
Sum of end-to-end Manhattan distances over all self-avoiding walks on square lattice trapped after n steps.
Original entry on oeis.org
1, 4, 21, 72, 271, 858, 2846, 8632, 26913, 79504, 238881, 693210, 2033133, 5823100, 16794540, 47619222, 135663289, 381615476, 1077064799, 3010363236, 8434161574, 23448994128
Offset: 7
a(9)=21 because the A077482(9)=11 different self-trapping walk stop at 5*(0,1)->d=1, 2*(1,2)->d=3, 2*(2,1)->d=3,(-1,0)->d=1,(3,0)->d=3. a(9) = 5*1+2*3+2*3+1+3 = 21.
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