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

1,6

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

Table of n, a(n) for n=1..76.

Robin Houston, Symmetry of bicycle lock numbers.

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

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

Sequence in context: A117192 A078729 A300236 * A029906 A094907 A343401

Adjacent sequences: A238155 A238156 A238157 * A238159 A238160 A238161

Keyword

tabl,nonn

Author

Robin Houston, Mar 23 2014

Status

approved