A373889 Square array read by ascending antidiagonals: T(k,n) is the cardinality of {(E is a proper finite subset of the natural numbers) such that E = {} or w_k(E) < min(E) <= max(E) <= n}, where w_k(E) = Sum_{i in E, i <> k} 1, with n, k >= 1.

2, 1, 3, 1, 2, 4, 1, 2, 4, 6, 1, 2, 4, 7, 9, 1, 2, 3, 6, 11, 14, 1, 2, 3, 6, 10, 17, 22, 1, 2, 3, 5, 10, 17, 26, 35, 1, 2, 3, 5, 10, 16, 28, 40, 56, 1, 2, 3, 5, 8, 16, 26, 45, 62, 90, 1, 2, 3, 5, 8, 16, 26, 43, 71, 97, 145, 1, 2, 3, 5, 8, 13, 26, 42, 71, 111, 153, 234

Offset: 1

Author

Paolo Xausa, Jun 21 2024

Keywords

• nonn
• tabl

Examples

			The array begins:
  k\n|  1  2  3  4   5   6   7   8   9   10  ...
  ----------------------------------------------
   1 |  2, 3, 4, 6,  9, 14, 22, 35, 56,  90, ... = A001611 (from n = 2).
   2 |  1, 2, 4, 7, 11, 17, 26, 40, 62,  97, ...
   3 |  1, 2, 4, 6, 10, 17, 28, 45, 71, 111, ...
   4 |  1, 2, 3, 6, 10, 16, 26, 43, 71, 116, ...
   5 |  1, 2, 3, 5, 10, 16, 26, 42, 68, 111, ...
   6 |  1, 2, 3, 5,  8, 16, 26, 42, 68, 110, ...
   7 |  1, 2, 3, 5,  8, 13, 26, 42, 68, 110, ...
   8 |  1, 2, 3, 5,  8, 13, 21, 42, 68, 110, ...
   9 |  1, 2, 3, 5,  8, 13, 21, 34, 68, 110, ...
  10 |  1, 2, 3, 5,  8, 13, 21, 34, 55, 110, ...
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).
• Hung Viet Chu and Zachary Louis Vasseur, Weighted Schreier-type Sets and the Fibonacci Sequence, arXiv:2405.19352 [math.CO], 2024. See p. 2, Table 1 and Theorem 1.2.

Crossrefs

Cf. A000045, A001611, A373345, A373556, A373557.

Programs

Mathematica

A373889[k_, n_] := Which[n < k, Fibonacci[n+1], k == 1, Fibonacci[n-k+2] + 1, True, 2*Sum[Binomial[n-k, i]*Fibonacci[k-i], {i, 0, k-2}] + 2*Binomial[n-k, k-1] + Sum[Binomial[j, n-j], {j, n-k}]];
Table[A373889[k-n+1, n], {k, 15}, {n, k}]

Formula

T(k,n) = A000045(n-k+2) + 1, for k = 1 and n >= k;

T(k,n) = 2*(Sum_{i=0..k-2} binomial(n-k,i)*A000045(k-i)) + 2*binomial(n-k,k-1) + Sum_{j=1..n-k} binomial(j,n-j), for k >= 2 and n >= k;

T(k,n) = A000045(n+1) otherwise.

T(n,n) = 2*A000045(n).