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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.
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%I #60 Apr 26 2016 12:31:13

%S 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 15,26,62,62,182,16,1,1,15,24,126,781,126,42,12,1,1,511,1640,30,24,

%U 3751,114,28,24,1,1,63,9841,30,20,1638,2801,28,78,60,1

%N 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 For definition of Danzer matrix see [Jeffery] (notation differs there!).

%C 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 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 Definition. If T(n,k)>=(k^n-1)/(k-1), for some k>1, then T(n,k) is said to be "optimal."

%C Conjecture 3. If T(n,k) is optimal, then n is a Queneau number (A054639).

%C Sequence is read from antidiagonals of array T which begins as

%C .1...1....1....1......1.......1......1....1.....1.........1

%C .1...3....8....6.....20......24.....16...12....24........60

%C .1...7...26...14.....62.....182.....42...28....78.......434

%C .1...7...18...14.....62.....126....114...28....54.......434

%C .1..31..121...62....781....3751...2801..124...363.....24211

%C .1..63...26..126.....24....1638..13072..252....78.......504

%C .1..15...24...30.....20.....120....400...60....72........60

%C .1..15.1640...30..32552....4920.240200...60..4920....488280

%C .1.511.9841.1022.488281.5028751....342.2044.29523.249511591

%C .1..63...78..126....124....1638.....42..252...234......7812

%C 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 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 L. E. Jeffery, <a href="/A220555/a220555_2.pdf">Danzer matrices</a>.

%Y Cf. A001175 (possibly = row 2), A086839 (possibly = column 2), A160893, A160895, A160897, A160960, A160972, A161010, A161025, A161139, A161167, A161213.

%Y Cf. A187772 (gives maximal periods p of Conjecture 1).

%K nonn,hard,tabl

%O 1,5

%A _L. Edson Jeffery_, Dec 15 2012