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%I #88 May 22 2020 01:14:14
%S 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,
%T 7,342,125,4,0,0,0,9,80,0,0,7775,0,0,0,0,10,9,8,7
%N 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.
%C 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)|.
%C T(7,5) was not known in 1998 (Chalk).
%D 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 D. L. Johnson, Presentation of Groups, Cambridge, 1976, see table p. 182.
%D 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.
%D Nikolova, Daniela B., The Fibonacci groups - four years later, in Semigroups (Kunming, 1995), 251-255, Springer, Singapore, 1998.
%D Nikolova, D. B.; and Robertson, E. F., One more infinite Fibonacci group. C. R. Acad. Bulgare Sci. 46 (1993), no. 3, 13-15.
%D 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.
%H Brunner, A. M., <a href="http://dx.doi.org/10.1017/S0004972700043574">The determination of Fibonacci groups</a>, Bull. Austral. Math. Soc. 11 (1974), 11-14.
%H A. M. Brunner, <a href="https://doi.org/10.1017/S0013091500026286">On groups of Fibonacci type</a>, Proc. Edinburgh Math. Soc. (2) 20 (1976/77), no. 3, 211-213.
%H C. M. Campbell and P. P. Campbell, <a href="https://www.maths.tcd.ie/pub/ims/bull56/GiG5602.pdf">Search techniques and epimorphisms between certain groups and Fibonacci groups</a>, Irish Math. Soc. Bull. No. 56 (2005), 21-28.
%H Chalk, Christopher P., <a href="http://dx.doi.org/10.1080/00927879808826218"> Fibonacci groups with aspherical presentations</a>, Comm. Algebra 26 (1998), no. 5, 1511-1546.
%H C. P. Chalk and D. L. Johnson, <a href="https://doi.org/10.1017/S0308210500018059">The Fibonacci groups II</a>, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), no. 12, 79-86.
%H J. H. Conway et al., Advanced problem 5327, Amer. Math. Monthly, 72 (1965), <a href="http://www.jstor.org/stable/2315059">915</a>; 74 (1967), <a href="http://www.jstor.org/stable/2314082">91-93</a>.
%H Helling, H.; Kim, A. C.; and Mennicke, J. L.; <a href="https://www.emis.de/journals/JLT/vol.8_no.1/1.html">A geometric study of Fibonacci groups</a>, J. Lie Theory 8 (1998), no. 1, 1-23.
%H Derek F. Holt, <a href="https://projecteuclid.org/euclid.em/1047931620">An alternative proof that the Fibonacci group F(2,9) is infinite</a>, Experiment. Math. 4 (1995), no. 2, 97-100.
%H David J. Seal, <a href="https://doi.org/10.1017/S0308210500032479">The orders of the Fibonacci groups</a>, Proc. Roy. Soc. Edinburgh, Sect. A 92 (1982), no. 3-4, 181-192.
%H A. Szczepanski, <a href="https://doi.org/10.1093/qjmath/52.3.385">The Euclidean representations of the Fibonacci groups</a>, Quart. J. Math. 52 (2001), 385-389.
%e The array begins:
%e k = 1 2 3 4 5 6 7 8 9 10 ...
%e ----------------------------------------------------------
%e n=1: 0 0 0 0 0 0 0 0 0 0 ...
%e n=2: 1 1 8 5 11 0 29 0 0 0 ...
%e n=3: 2 8 2 0 22 1512 0 0 0 0 ...
%e n=4: 3 3 63 3 0 0 0 0 ? 0 ...
%e n=5: 4 24 0 624 4 0 0 0 0 0 ...
%e n=6: 5 5 5 125 7775 5 0 0 0 0 ...
%e n=7: 6 48 342 0 ? 7^6-1 6 0 0 0 ...
%e n=8: 7 7 0 7 ? 0 8^7-1 7 0 0 ...
%e n=9: 8 80 8 6560 0 0 0 9^8-1 8 0 ...
%e n=10 9 9 999 4905 9 ? ? 0 10^9-1 9 ...
%e ...
%e 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.
%e 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.
%Y Cf. A037205 (a diagonal), A065530, A202625, A202626, A202627 (columns).
%K nonn,tabl,more,nice
%O 2,2
%A _N. J. A. Sloane_, Dec 29 2011
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