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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
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.
LINKS
FORMULA
T(n,k) = max { min{xy, (n-x)(k-y)} | 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}]]
CROSSREFS
Sequence in context: A117192 A078729 A300236 * A029906 A094907 A343401
KEYWORD
tabl,nonn
AUTHOR
Robin Houston, Mar 23 2014
STATUS
approved

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Last modified July 21 16:35 EDT 2024. Contains 374475 sequences. (Running on oeis4.)