[seqfan] 3-peg Tower of Hanoi seqs

Richard J. Mathar mathar at mpia-hd.mpg.de
Fri Jul 8 16:55:02 CEST 2022

```There are apparently 4 new sequences in
Amir Sapir, The Tower of Hanoi with Forbidden Moves, The Computer J. 47 (1) (2004) 20
for the case "(v) complete--", three pegs in a cyclic arrangement:

a(n>=0):0,1,4,11,28,67,160,375,884,2067,4856,11359,26652,62379,146272,342487,802820,1880227,4406536,10321807,24187500
g.f.-x*(1+2*x)/(x-1)/(2*x^3-4*x^2-x+1)

b(n>=0):0,2,7,19,47,113,267,629,1475,3461,8107,19005,44515,104325,244379,572653,1341523,3143381,7364171,17254653,40424579
g.f. x*(-2-3*x+x^2)/(x-1)/(2*x^3-4*x^2-x+1).

c(n>=0):0,1,3,8,19,46,107,254,591,1394,3251,7646,17863,41946,98107,230166,538703,1263154,2957635,6932846,16237079
g.f. x*(-1-x+x^2)/(x-1)/(2*x^3-4*x^2-x+1).

d(n>=0):0,1,3,7,17,39,93,215,509,1183,2789,6503,15293,35727,83893,196215,460333,1077407,2526309,5915271,13865693
g.f. x*(1+2*x)/(1-x-4*x^2+2*x^3) .

a(n)=b(n-1)+c(n-1)+1 ;
b(n)=2*b(n-1)+d(n-1)+2 ;
c(n)=a(n-1)+d(n-1)+1;
d(n)=2*c(n-1)+1.

Two other sequences of the article for the case "(iv) Cyclic++" appear in A026583 and A026599.

```