Here we consider the set S(n;t,s) of length n words a₁a₂···a_n over the alphabet consisting of the elements of the field F₅ that have trace t and subtrace s. The trace of a word is the sum of its digits over the field F₅ , i.e. t = a₁+a₂+ ··· +a_n. The subtrace is the sum of the products of all n(n-1)/2 pairs of digits taken over the field F₅ , i.e. s = SUM( a_ia_j : 1 < i < j < n ). Note that F₅ = Z₅ .

Examples:

• The two 5-ary strings of trace 0, subtrace 4 and length 2 are { 14, 41 }.
• The two 5-ary strings of trace 3, subtrace 2 and length 2 are { 12, 21 }.
• The one 5-ary string of trace 1, subtrace 2 and length 3 is { 222 }.

Further Notes:

• The number S(n;t,s) can be computed from the following recurrence relation

  S(n;t,s) = S(n-1;t,s) + S(n-1;t-1,s-(t-1)) + S(n-1;t-2,s-2(t-2)) + S(n-1;t-3,s-3(t-3)) + S(n-1;t-4,s-4(t-4))

  Note that all operations involving operands t or s are carried out over GF(5).

• Column (0,0) is sequence A073963 in Neil J. Sloane's database of integer sequences.
• Column (0,1),(0,4) is sequence A073964 in Neil J. Sloane's database of integer sequences.
• Column (0,2),(0,3) is sequence A073965 in Neil J. Sloane's database of integer sequences.
• Column (1,0),(2,0),(3,0),(4,0) is sequence A073966 in Neil J. Sloane's database of integer sequences.
• Column (1,1),(2,4),(3,4),(4,1) is sequence A073967 in Neil J. Sloane's database of integer sequences.
• Column (1,2),(2,3),(3,3),(4,2) is sequence A073968 in Neil J. Sloane's database of integer sequences.
• Column (1,3),(2,2),(3,2),(4,3) is sequence A073969 in Neil J. Sloane's database of integer sequences.
• Column (1,4),(2,1),(3,1),(4,4) is sequence A073970 in Neil J. Sloane's database of integer sequences.

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