A381476 Triangle read by rows: T(n,k) is the number of subsets of {1..n} with k elements such that every pair of distinct elements has a different difference, 0 <= k <= A143824(n).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 4, 6, 2, 1, 5, 10, 6, 1, 6, 15, 14, 1, 7, 21, 26, 2, 1, 8, 28, 44, 10, 1, 9, 36, 68, 26, 1, 10, 45, 100, 60, 1, 11, 55, 140, 110, 1, 12, 66, 190, 190, 4, 1, 13, 78, 250, 304, 22, 1, 14, 91, 322, 466, 68, 1, 15, 105, 406, 676, 156

Offset: 0

Author

Andrew Howroyd, Mar 27 2025

Keywords

• nonn
• tabf

Comments

Equivalently, a(n) is the number of Sidon sets of {1..n} of size k.

Examples

			Triangle begins:
   0 | 1;
   1 | 1,  1;
   2 | 1,  2,  1;
   3 | 1,  3,  3;
   4 | 1,  4,  6,   2;
   5 | 1,  5, 10,   6;
   6 | 1,  6, 15,  14;
   7 | 1,  7, 21,  26,   2;
   8 | 1,  8, 28,  44,  10;
   9 | 1,  9, 36,  68,  26;
  10 | 1, 10, 45, 100,  60;
  11 | 1, 11, 55, 140, 110;
  12 | 1, 12, 66, 190, 190, 4;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..464 (rows 0..60)
• Wikipedia, Sidon sequence.
• Index entries for sequences related to Golomb rulers.

Crossrefs

Columns 0..5 are A000012, A001477, A161680, A212964(n-1), A241688, A241689, A241690.

Row sums are A143823.

Cf. A003022, A143824, A325879, A382395.

Programs

PARI

row(n)={
  local(L=List());
  my(recurse(k,r,b,w)=
      if(k > n, if(r>=#L,listput(L,0)); L[1+r]++,
         self()(k+1, r, b, w);
         b+=1<
				

Formula

T(n,A143824(n)) = A382395(n).