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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.

%I A238158 #36 May 14 2021 08:12:19
%S A238158 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,
%T A238158 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,
%U A238158 12,15,18,20,24,25,30,0,6,8,12,14,18,20,24,25,30
%N 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.
%C A238158 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.
%C A238158 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.
%C A238158 T(n, k) <= nk/4, with equality when n and k are both even.
%H A238158 Robin Houston, <a href="http://math.stackexchange.com/questions/683711/symmetry-of-bicycle-lock-numbers/689228#689228">Symmetry of bicycle lock numbers</a>.
%F A238158 T(n,k) = max { min{xy, (n-x)(k-y)} | 0<=x<=n, 0<=y<=k; x, y integers }.
%e A238158 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:
%e A238158 0
%e A238158 0 1
%e A238158 0 1 2
%e A238158 0 2 2 4
%e A238158 0 2 3 4 6
%e A238158 0 3 4 6 6 9
%e A238158 0 3 4 6 8 9 12
%e A238158 0 4 5 8 9 12 12 16
%e A238158 0 4 6 8 10 12 15 16 20
%e A238158 0 5 6 10 12 15 16 20 20 25
%e A238158 0 5 7 10 12 15 18 20 24 25 30
%e A238158 0 6 8 12 14 18 20 24 25 30 30 36
%e A238158 0 6 8 12 15 18 21 24 28 30 35 36 42
%t A238158 t[a_, b_] := Max[Table[Min[x*y, (a - x)*(b - y)], {x, 0, a}, {y, 0, b}]]
%Y A238158 Cf. A000004, A004526, A004523, A052928, A239492.
%K A238158 tabl,nonn
%O A238158 1,6
%A A238158 _Robin Houston_, Mar 23 2014