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A361928
Triangle read by rows: T(n,d) = number of non-adaptive group tests required to identify exactly d defectives among n items.
0
1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 3, 5, 5, 5, 5, 3, 6, 6, 6, 6, 6, 3, 6, 7, 7, 7, 7, 7, 4, 7, 8, 8, 8, 8, 8, 8, 4, 7, 9, 9, 9, 9, 9, 9, 9, 4, 8, 10, 10, 10, 10, 10, 10, 10, 10, 4, 8, 11, 11, 11, 11, 11, 11, 11, 11, 11, 4, 8, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 4, 9, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 4, 9, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 4, 9
OFFSET
2,2
COMMENTS
Arguably, the triangle should be flanked by zeros on both sides -- for all n, T(n,0)=0 and T(n,n)=0 -- but these are not included here.
T(n,d) is the smallest number of rows that an n-column matrix can have while remaining d-separable.
Observations:
T(n,n-1) = n-1.
T(n,d) <= T(n+1,d).
T(n,d) <= T(n,d+1) whenever d <= n-2.
T(n,d) <= T(n+1,d) <= T(n,d)+1 whenever d < n.
T(n,d) < T(n+1,d+1) whenever d < n.
T(n+2,k+1) = n+1 whenever T(n,k)=n-1.
T(n, ceil(n/2)) = n-1 for all n >= 1.
Elaqqad writes (see Links): "The only value of d for which T(n,d) is known completely is d=1, for which T(n,1)=ceil(lg n); the exact value even for d=2 is not known. Generally, the solutions to this problem are called 'testing designs', and the main considered ones are: (1) Set-packing designs or block designs; (2) Transversal designs; (3) Designs whose d-disjunct or d-separable matrices are directly constructed."
EXAMPLE
Triangle begins:
1;
2, 2;
2, 3, 3;
3, 4, 4, 4;
3, 5, 5, 5, 5;
3, 6, 6, 6, 6, 6;
3, 6, 7, 7, 7, 7, 7;
4, 7, 8, 8, 8, 8, 8, 8;
4, 7, 9, 9, 9, 9, 9, 9, 9;
4, 8, 10, 10, 10, 10, 10, 10, 10, 10;
...
If we have 8 items, 3 of which are defective, we can identify the 3 defectives in 6 tests:
Test 1. T..TT...
Test 2. T....TT.
Test 3. .T.T.T..
Test 4. .T..T.T.
Test 5. ..T.TT..
Test 6. ..TT..T.
For example: If tests (1,2,3,4,5) are positive, then items (1,2,5) are the defectives. If tests (2,3,4,5,6) are positive, then items (6,7,8) are the defectives. If tests (2,4,5,6) are positive, then items (3,7,8) are the defectives.
CROSSREFS
Cf A054961: A054961(i) is the smallest n such that T(n,2)=i.
Cf A290492: A290492(i) is the smallest n such that T(n,3)=i.
Sequence in context: A122651 A343378 A351700 * A300013 A130535 A329194
KEYWORD
nonn,tabl
AUTHOR
Arthur O'Dwyer, Mar 30 2023
STATUS
approved