A305499 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.
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, 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, 1, 2, 2, 3, 3, 5, 6, 8, 9, 11, 12, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 10, 11, 13
Offset: 1
Examples
The square array begins:
n\k | 1 2 3 4 5 6 7 8 9 10 11 12
====+=======================================
1 | 1 2 3 4 5 6 7 8 9 10 11 12
2 | 1 1 3 3 5 5 7 7 9 9 11 11
3 | 1 1 2 3 4 5 6 7 8 9 10 11
4 | 1 1 2 2 4 4 6 6 8 8 10 10
5 | 1 1 2 2 3 4 5 6 7 8 9 10
6 | 1 1 2 2 3 3 5 5 7 7 9 9
7 | 1 1 2 2 3 3 4 5 6 7 8 9
8 | 1 1 2 2 3 3 4 4 6 6 8 8
9 | 1 1 2 2 3 3 4 4 5 6 7 8
10 | 1 1 2 2 3 3 4 4 5 5 7 7
11 | 1 1 2 2 3 3 4 4 5 5 6 7
etc.
Crossrefs
Formula
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.
A(n+m,n) = floor((n+1)/2) for n > 0 and some fixed m >= 0.
A(n,n+m) = floor((m+1)/2) + floor((n+1+m)/2) for n>0 and some fixed m >= 0.
A(n+1,k+1) = A(n,k+1) + A(n,k) - A(n-1,k) for k > 0 and n > 1.
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.
A(n,n) = A008619(n-1) for n > 0.
A(n+1,2*n-1) = A001651(n) for n > 0.
Sum_{i=1..n} A(i,i)*A209229(i) = 2^floor(log_2(n)) for n > 0.
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.
P(n+1,x) = P(n,x) - x^n/(1-x^2) for n > 0 and P(1,x) = 1/(1-x)^2.
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)).
Conjecture: Sum_{i=1..n} A(n+1-i,i) = A211538(n+3) for n > 0.