A239492 -id:A239492 - cp's OEIS frontend

A238158 Bicycle lock numbers: triangle T(n,k) with 1<=k<=n, where T(n,k) is the maximum value of min{xy, (n-x)(k-y)} over 0 <= x <= n, 0 <= y <= k for integers x, y.

0, 0, 1, 0, 1, 2, 0, 2, 2, 4, 0, 2, 3, 4, 6, 0, 3, 4, 6, 6, 9, 0, 3, 4, 6, 8, 9, 12, 0, 4, 5, 8, 9, 12, 12, 16, 0, 4, 6, 8, 10, 12, 15, 16, 20, 0, 5, 6, 10, 12, 15, 16, 20, 20, 25, 0, 5, 7, 10, 12, 15, 18, 20, 24, 25, 30, 0, 6, 8, 12, 14, 18, 20, 24, 25, 30

Offset: 1

Robin Houston, Mar 23 2014

Really an infinite symmetric matrix: the definition is symmetric in n and k. As a symmetric matrix, the first few rows are: A000004, A004526, A004523, A052928, A239492.
T(n+1, k) is the minimum number of turns that always suffice to open from any starting position a bicycle lock that has n dials with k numbers on each dial, where a turn consists of simultaneously rotating any number of adjacent dials by one place.
T(n, k) <= nk/4, with equality when n and k are both even.

			For n=5, k=4, the maximum value is attained at x=2, y=2, so T(5, 4) = 2*2 = 4. The first few rows of the triangle are:
0
0 1
0 1 2
0 2 2 4
0 2 3 4 6
0 3 4 6 6 9
0 3 4 6 8 9 12
0 4 5 8 9 12 12 16
0 4 6 8 10 12 15 16 20
0 5 6 10 12 15 16 20 20 25
0 5 7 10 12 15 18 20 24 25 30
0 6 8 12 14 18 20 24 25 30 30 36
0 6 8 12 15 18 21 24 28 30 35 36 42

Robin Houston, Symmetry of bicycle lock numbers.

Cf. A000004, A004526, A004523, A052928, A239492.

t[a_, b_] := Max[Table[Min[x*y, (a - x)*(b - y)], {x, 0, a}, {y, 0, b}]]

T(n,k) = max { min{xy, (n-x)(k-y)} | 0<=x<=n, 0<=y<=k; x, y integers }.

cp's OEIS Frontend

A238158 Bicycle lock numbers: triangle T(n,k) with 1<=k<=n, where T(n,k) is the maximum value of min{xy, (n-x)(k-y)} over 0 <= x <= n, 0 <= y <= k for integers x, y.

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