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

1,3

Links

Table of n, a(n) for n=1..91.

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.

Example

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

Cf. A000012 (col 1), A054977 (col 2), A000027 (row 1), A109613 (row 2), A028310 (row 3), A008619 (main diagonal and subdiagonals).

Cf. A000035, A001651, A209229, A211538.

Sequence in context: A371209 A353947 A181846 * A210873 A224838 A030272

Adjacent sequences: A305496 A305497 A305498 * A305500 A305501 A305502

Keyword

nonn,tabl,easy

Author

Werner Schulte, Jun 03 2018

Status

approved