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Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of coprime pairs (a,b) with -n <= a <= n, -k <= b <= k.
3

%I #27 Nov 29 2017 17:00:02

%S 8,12,12,16,16,16,20,24,24,20,24,28,32,28,24,28,36,40,40,36,28,32,40,

%T 52,48,52,40,32,36,48,56,64,64,56,48,36,40,52,68,68,80,68,68,52,40,44,

%U 60,76,84,88,88,84,76,60,44,48,64,84,92,108,96,108,92,84,64,48

%N Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of coprime pairs (a,b) with -n <= a <= n, -k <= b <= k.

%H Seiichi Manyama, <a href="/A295782/b295782.txt">Antidiagonals n = 1..140, flattened</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Coprime_integers">Coprime integers</a>

%F A(n,k) = A(k,n).

%e A(2,1)=12 because there are twelve coprime pairs (1,0), (2,1), (1,1), (0,1), (-1,1), (-2,1), (-1,0), (-2,-1), (-1,-1), (0,-1), (1,-1), (2,-1).

%e Square array begins:

%e 8, 12, 16, 20, 24, 28, 32, ...

%e 12, 16, 24, 28, 36, 40, 48, ...

%e 16, 24, 32, 40, 52, 56, 68, ...

%e 20, 28, 40, 48, 64, 68, 84, ...

%e 24, 36, 52, 64, 80, 88, 108, ...

%e 28, 40, 56, 68, 88, 96, 120, ...

%e 32, 48, 68, 84, 108, 120, 144, ...

%e 36, 52, 76, 92, 120, 132, 160, ...

%Y For n > 0, columns k = 2,3 give A295821, A295822.

%Y Main diagonal gives A137243.

%K nonn,tabl

%O 1,1

%A _Seiichi Manyama_, Nov 27 2017