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A220555 T(n,k) = maximal order N of cyclic group {D,D^2,...,D^N} generated by an n X n Danzer matrix D over Z/kZ, where D is from the m-th Danzer basis and m=2*n+1.

%I A220555 #60 Apr 26 2016 12:31:13
%S A220555 1,1,1,1,3,1,1,7,8,1,1,7,26,6,1,1,31,18,14,20,1,1,63,121,14,62,24,1,1,
%T A220555 15,26,62,62,182,16,1,1,15,24,126,781,126,42,12,1,1,511,1640,30,24,
%U A220555 3751,114,28,24,1,1,63,9841,30,20,1638,2801,28,78,60,1
%N A220555 T(n,k) = maximal order N of cyclic group {D,D^2,...,D^N} generated by an n X n Danzer matrix D over Z/kZ, where D is from the m-th Danzer basis and m=2*n+1.
%C A220555 For definition of Danzer matrix see [Jeffery] (notation differs there!).
%C A220555 Conjecture 1. Let F_n(x)=sum_{j=0..n} A187660(n,j)*x^{(n-1)*j}. Let f_n in Z[x] be any polynomial in x of degree d such that 0<=d<=(n-1)*(n-2). Then the sequence of coefficients of the series expansion of f_n(x)/F_n(x), when taken over Z/kZ, is periodic with period p <= (n-1)*A220555(n,k), for all n,k > 1. (Cf. [Coleman, et al.] for the case for n=2 (generalized Fibonacci).)
%C A220555 Conjecture 2. If G a cyclic multiplicative group generated by an n X n integer matrix over Z/kZ, then |G|<=T(r,k), for some r<=n.
%C A220555 Definition. If T(n,k)>=(k^n-1)/(k-1), for some k>1, then T(n,k) is said to be "optimal."
%C A220555 Conjecture 3. If T(n,k) is optimal, then n is a Queneau number (A054639).
%C A220555 Sequence is read from antidiagonals of array T which begins as
%C A220555 .1...1....1....1......1.......1......1....1.....1.........1
%C A220555 .1...3....8....6.....20......24.....16...12....24........60
%C A220555 .1...7...26...14.....62.....182.....42...28....78.......434
%C A220555 .1...7...18...14.....62.....126....114...28....54.......434
%C A220555 .1..31..121...62....781....3751...2801..124...363.....24211
%C A220555 .1..63...26..126.....24....1638..13072..252....78.......504
%C A220555 .1..15...24...30.....20.....120....400...60....72........60
%C A220555 .1..15.1640...30..32552....4920.240200...60..4920....488280
%C A220555 .1.511.9841.1022.488281.5028751....342.2044.29523.249511591
%C A220555 .1..63...78..126....124....1638.....42..252...234......7812
%C A220555 Rows might be related to Jordan totient functions J_n(k), however, some entries T(n,k) are products of factors of the form (j^n-1)/(j-1).
%H A220555 D. A. Coleman et al., <a href="http://webbox.lafayette.edu/~reiterc/nt/qr_fib_ec_preprint.pdf">Periods of (q,r)-Fibonacci sequences and Elliptic Curves</a>, Fibonacci Quart. 44, no 1 (2006) 59-70.
%H A220555 L. E. Jeffery, <a href="/A220555/a220555_2.pdf">Danzer matrices</a>.
%Y A220555 Cf. A001175 (possibly = row 2), A086839 (possibly = column 2), A160893, A160895, A160897, A160960, A160972, A161010, A161025, A161139, A161167, A161213.
%Y A220555 Cf. A187772 (gives maximal periods p of Conjecture 1).
%K A220555 nonn,hard,tabl
%O A220555 1,5
%A A220555 _L. Edson Jeffery_, Dec 15 2012