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%I #13 Feb 07 2020 20:49:14
%S 1,1,2,1,1,3,1,1,3,4,1,1,2,3,5,1,1,2,3,5,6,1,1,2,2,4,5,7,1,1,2,2,4,5,
%T 7,8,1,1,2,2,3,4,6,7,9,1,1,2,2,3,4,6,7,9,10,1,1,2,2,3,3,5,6,8,9,11,1,
%U 1,2,2,3,3,5,6,8,9,11,12,1,1,2,2,3,3,4,5,7,8,10,11,13
%N Square array A(n,k), n > 0 and k > 0, read by antidiagonals, with initial values A(1,k) = k and recurrence equations A(n+1,k) = A(n,k) for 0 < k <= n and A(n+1,k) = A(n,k) - A000035(n+k) for 0 < n < k.
%F A(n,k) = floor((k+1)/2) for 1 <= k <= n and A(n,k) = floor((k+1)/2) + floor((k+1-n)/2) for 1 <= n < k.
%F A(n+m,n) = floor((n+1)/2) for n > 0 and some fixed m >= 0.
%F A(n,n+m) = floor((m+1)/2) + floor((n+1+m)/2) for n>0 and some fixed m >= 0.
%F A(n+1,k+1) = A(n,k+1) + A(n,k) - A(n-1,k) for k > 0 and n > 1.
%F A(n,k) = A(n,k-1) + 2*A(n,k-2) - 2*A(n,k-3) - A(n,k-4) + A(n,k-5) for n > 0 and k > 5.
%F A(n,n) = A008619(n-1) for n > 0.
%F A(n+1,2*n-1) = A001651(n) for n > 0.
%F Sum_{i=1..n} A(i,i)*A209229(i) = 2^floor(log_2(n)) for n > 0.
%F P(n,x) = Sum_{k>0} A(n,k)*x^(k-1) = (1-x^(2*n))/((1-x^n)*(1-x^2)*(1-x)) = (1+x^n)/((1-x^2)*(1-x)) for n > 0.
%F P(n+1,x) = P(n,x) - x^n/(1-x^2) for n > 0 and P(1,x) = 1/(1-x)^2.
%F G.f.: Sum_{n>0, k>0} A(n,k)*x^(k-1)*y^(n-1) = (1+x-2*x*y)/((1-x)*(1-x^2) * (1-y)*(1-x*y)).
%F Conjecture: Sum_{i=1..n} A(n+1-i,i) = A211538(n+3) for n > 0.
%e The square array begins:
%e n\k | 1 2 3 4 5 6 7 8 9 10 11 12
%e ====+=======================================
%e 1 | 1 2 3 4 5 6 7 8 9 10 11 12
%e 2 | 1 1 3 3 5 5 7 7 9 9 11 11
%e 3 | 1 1 2 3 4 5 6 7 8 9 10 11
%e 4 | 1 1 2 2 4 4 6 6 8 8 10 10
%e 5 | 1 1 2 2 3 4 5 6 7 8 9 10
%e 6 | 1 1 2 2 3 3 5 5 7 7 9 9
%e 7 | 1 1 2 2 3 3 4 5 6 7 8 9
%e 8 | 1 1 2 2 3 3 4 4 6 6 8 8
%e 9 | 1 1 2 2 3 3 4 4 5 6 7 8
%e 10 | 1 1 2 2 3 3 4 4 5 5 7 7
%e 11 | 1 1 2 2 3 3 4 4 5 5 6 7
%e etc.
%Y Cf. A000012 (col 1), A054977 (col 2), A000027 (row 1), A109613 (row 2), A028310 (row 3), A008619 (main diagonal and subdiagonals).
%Y Cf. A000035, A001651, A209229, A211538.
%K nonn,tabl,easy
%O 1,3
%A _Werner Schulte_, Jun 03 2018
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