A077818 a(n) is the numerator of the probability P(n) of the occurrence of a 3-dimensional self-trapping walk of length n.

40, 190, 15925, 48795, 86221819, 28522360751, 583791967829, 1801511107253, 32337280749408865

Offset: 11

Author

Hugo Pfoertner, Nov 17 2002

Keywords

• nonn
• frac
• walk
• hard
• more

Comments

A comparison of the exact probabilities with simulation results obtained for 1.1*10^9 random walks is given under "Results of simulation, comparison with exact probabilities" in the first link. The behavior of P(n) for larger values of n is illustrated in "Probability density for the number of steps before trapping occurs" at the same location. P(n) has a maximum around n~=600 (P(600)~=1/4760) and drops exponentially for large n (P(45000)~=1/10^9). The average walk length determined by the numerical simulation is sum n=11..infinity (n*P(n))=3953.65 +-0.20.

A more accurate value for this length, determined from a simulation with 27*10^9 walks, is 3953.8+-0.1 (A378903). - Hugo Pfoertner, Dec 15 2024

Examples

			a(13)=15925, A077819(13)=A077820(13)=1 because there are 5 different probabilities for the 1832 (=8*A077817(13)) walks: 256 walks with probability p1=1/125000000, 88 with p2=1/146484375, 600 with p3=1/156250000, 728 with p4=1/146484375 and 160 with p5=1/244140625. P(13)=256*p1+88*p2+600*p3+728*p4+160*p5=637/(6*5^10)=25*637/(5^12*6)= 15295/(5^(13-1)*3^1*2^1)
		

References

• See under A001412.
• More references are given in the sci.math NG posting in the second link.

Links

• Hugo Pfoertner, Results for the 3-dimensional Self-Trapping Random Walk.
• Hugo Pfoertner, Self-trapping random walks on square lattice in 2-D (cubic in 3-D). Posting in NG sci.math dated March 5, 2002.
• Alexander Renner, Self avoiding walks and lattice polymers, Diplomarbeit, Universität Wien, December 1994.

Crossrefs

Cf. A001412, A077817, A077819, A077820.

Cf. A377161, A377162, A378903.

Programs

Fortran

c Program provided at first link

Formula

P(n) = a(n) / (5^(n-1) * 3^A077819(n) * 2^A077820(n)) = A377161(n)/A377162(n).