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.

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, 15, 26, 62, 62, 182, 16, 1, 1, 15, 24, 126, 781, 126, 42, 12, 1, 1, 511, 1640, 30, 24, 3751, 114, 28, 24, 1, 1, 63, 9841, 30, 20, 1638, 2801, 28, 78, 60, 1

Offset: 1

Author

L. Edson Jeffery, Dec 15 2012

Keywords

• nonn
• hard
• tabl

Comments

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

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).)

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.

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

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

Sequence is read from antidiagonals of array T which begins as

.1...1....1....1......1.......1......1....1.....1.........1

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

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

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

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

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

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

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

.1.511.9841.1022.488281.5028751....342.2044.29523.249511591

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

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).

Links

• D. A. Coleman et al., Periods of (q,r)-Fibonacci sequences and Elliptic Curves, Fibonacci Quart. 44, no 1 (2006) 59-70.
• L. E. Jeffery, Danzer matrices.

Crossrefs

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

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