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Revision History for A220555
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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.
(history; published version)
(history; published version)
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KEYWORD
nonn,hard,tabl
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COMMENTS
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.] which deals with for the case for n=2 (generalized case for n=2 (Fibonacci).)
Conjecture 2. If G a cyclic multiplicative group generated by an n X n integer matrix g over Z/kZ, then |G|<=T(n,r,k), for some r<=n.
Conjecture 3. If T(n,k) is optimal, then n is a Queneau number (Cf. A054639).
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Conjecture 2. If G a cyclic multiplicative group generated by an n X n integer matrix g over Z/kZ, then |G|<=T(r,n,k), where r=rank(g).
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editing
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proposed
approved