A202624 Array read by antidiagonals: T(n,k) = order of Fibonacci group F(n,k), writing 0 if the group is infinite, for n >= 2, k >= 1.

1, 2, 1, 3, 8, 8, 4, 3, 2, 5, 5, 24, 63, 0, 11, 6, 5, 0, 3, 22, 0, 7, 48, 5, 624, 0, 1512, 29, 8, 7, 342, 125, 4, 0, 0, 0, 9, 80, 0, 0, 7775, 0, 0, 0, 0, 10, 9, 8, 7

Offset

2,2

Comments

The Fibonacci group F(r,n) has presentation <a_1,a_2,...,a_n|a_1*a_2*...*a_r=a_{r+1},...>, where there are n relations, obtained from the first relation by applying the permutation (1,2,,n) to the subscripts and reducing subscripts mod n. Then T(n,k) = |F(n,k)|.

T(7,5) was not known in 1998 (Chalk).

References

Campbell, Colin M.; and Gill, David M. On the infiniteness of the Fibonacci group F(5,7). Algebra Colloq. 3 (1996), no. 3, 283-284.

D. L. Johnson, Presentation of Groups, Cambridge, 1976, see table p. 182.

Mednykh, Alexander; and Vesnin, Andrei; On the Fibonacci groups, the Turk's head links and hyperbolic 3-manifolds, in Groups-Korea '94 (Pusan), 231-239, de Gruyter, Berlin, 1995.

Nikolova, Daniela B., The Fibonacci groups - four years later, in Semigroups (Kunming, 1995), 251-255, Springer, Singapore, 1998.

Nikolova, D. B.; and Robertson, E. F., One more infinite Fibonacci group. C. R. Acad. Bulgare Sci. 46 (1993), no. 3, 13-15.

Thomas, Richard M., The Fibonacci groups revisited, in Groups - St. Andrews 1989, Vol. 2, 445-454, London Math. Soc. Lecture Note Ser., 160, Cambridge Univ. Press, Cambridge, 1991.

Links

Table of n, a(n) for n=2..50.

Brunner, A. M., The determination of Fibonacci groups, Bull. Austral. Math. Soc. 11 (1974), 11-14.

A. M. Brunner, On groups of Fibonacci type, Proc. Edinburgh Math. Soc. (2) 20 (1976/77), no. 3, 211-213.

C. M. Campbell and P. P. Campbell, Search techniques and epimorphisms between certain groups and Fibonacci groups, Irish Math. Soc. Bull. No. 56 (2005), 21-28.

Chalk, Christopher P., Fibonacci groups with aspherical presentations, Comm. Algebra 26 (1998), no. 5, 1511-1546.

C. P. Chalk and D. L. Johnson, The Fibonacci groups II, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), no. 12, 79-86.

J. H. Conway et al., Advanced problem 5327, Amer. Math. Monthly, 72 (1965), 915; 74 (1967), 91-93.

Helling, H.; Kim, A. C.; and Mennicke, J. L.; A geometric study of Fibonacci groups, J. Lie Theory 8 (1998), no. 1, 1-23.

Derek F. Holt, An alternative proof that the Fibonacci group F(2,9) is infinite, Experiment. Math. 4 (1995), no. 2, 97-100.

David J. Seal, The orders of the Fibonacci groups, Proc. Roy. Soc. Edinburgh, Sect. A 92 (1982), no. 3-4, 181-192.

A. Szczepanski, The Euclidean representations of the Fibonacci groups, Quart. J. Math. 52 (2001), 385-389.

Example

The array begins:
k =  1  2   3    4    5     6     7     8      9    10 ...
----------------------------------------------------------
n=1: 0  0   0    0    0     0     0     0      0     0 ...
n=2: 1  1   8    5   11     0    29     0      0     0 ...
n=3: 2  8   2    0   22  1512     0     0      0     0 ...
n=4: 3  3  63    3    0     0     0     0      ?     0 ...
n=5: 4 24   0  624    4     0     0     0      0     0 ...
n=6: 5  5   5  125 7775     5     0     0      0     0 ...
n=7: 6 48 342    0    ? 7^6-1     6     0      0     0 ...
n=8: 7  7   0    7    ?     0 8^7-1     7      0     0 ...
n=9: 8 80   8 6560    0     0     0 9^8-1      8     0 ...
n=10 9  9 999 4905    9     ?     ?     0 10^9-1     9 ...
...
For example, T(2,5) = 11, since the presentation <a,b,c,d,e | ab=c, bc=d, cd=e, de=a, ea=b> defines the cyclic group of order 11. This example is due to John Conway.
This table is based on those in Johnson (1976) and Thomas (1989), supplemented by values from Chalk (1998). We have ignored the n=1 row when reading the table by antidiagonals.

Crossrefs

Cf. A037205 (a diagonal), A065530, A202625, A202626, A202627 (columns).

Sequence in context: A205391 A352858 A078045 * A342224 A145490 A329180

Adjacent sequences: A202621 A202622 A202623 * A202625 A202626 A202627

Keyword

nonn,tabl,more,nice

Author

N. J. A. Sloane, Dec 29 2011

Status

approved