

A238158


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


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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
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OFFSET

1,6


COMMENTS

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.


LINKS



FORMULA

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


EXAMPLE

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


MATHEMATICA

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


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KEYWORD



AUTHOR



STATUS

approved



