A096858 Triangle read by rows in which row n gives the n-set obtained as the differences {b(n)-b(n-i), 0 <= i <= n-1}, where b() = A005318().

1, 1, 2, 2, 3, 4, 3, 5, 6, 7, 6, 9, 11, 12, 13, 11, 17, 20, 22, 23, 24, 20, 31, 37, 40, 42, 43, 44, 40, 60, 71, 77, 80, 82, 83, 84, 77, 117, 137, 148, 154, 157, 159, 160, 161, 148, 225, 265, 285, 296, 302, 305, 307, 308, 309, 285, 433, 510, 550, 570, 581, 587, 590, 592, 593, 594

Offset: 1

Author

N. J. A. Sloane, Aug 18 2004

Keywords

• nonn
• tabl

Comments

It is conjectured that the triangle has the property that all 2^n subsets of row n have distinct sums. This conjecture was proved by T. Bohman in 1996 - N. J. A. Sloane, Feb 09 2012

It is also conjectured that in some sense this triangle is optimal. See A005318 for further information and additional references.

Examples

			The triangle begins:
{1}
{1,2}
{2,3,4}
{3,5,6,7}
{6,9,11,12,13}
{11,17,20,22,23,24}
{20,31,37,40,42,43,44}
{40,60,71,77,80,82,83,84}
{77,117,137,148,154,157,159,160,161}
{148,225,265,285,296,302,305,307,308,309}
{285,433,510,550,570,581,587,590,592,593,594}
{570,855,1003,1080,1120,1140,1151,1157,1160,1162,1163,1164}
{1120,1690,1975,2123,2200,2240,2260,2271,2277,2280,2282,2283,2284}
{2200,3320,3890,4175,4323,4400,4440,4460,4471,4477,4480,4482,4483,4484}
{4323,6523,7643,8213,8498,8646,8723,8763,8783,8794,8800,8803,8805,8806,8807}
		

References

• J. H. Conway and R. K. Guy, Solution of a problem of Erdős, Colloq. Math. 20 (1969), p. 307.
• R. K. Guy, Sets of integers whose subsets have distinct sums, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982.
• R. K. Guy, Unsolved Problems in Number Theory, C8.

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Tom Bohman, A sum packing problem of Erdős and the Conway-Guy sequence, Proc. AMS 124, (No. 12, 1996), pp. 3627-3636.

Crossrefs

Cf. A005318, A005230 (column 1 of triangle).

Programs

Maple

b:= proc(n) option remember;
      `if`(n<2, n, 2*b(n-1) -b(n-1-floor(1/2 +sqrt(2*n-2))))
    end:
T:= n-> seq(b(n)-b(n-i), i=1..n):
seq(T(n), n=1..15);  # Alois P. Heinz, Nov 29 2011

Mathematica

b[n_] := b[n] = If[n < 2, n, 2*b[n-1] - b[n-1-Floor[1/2 + Sqrt[2*n-2]]]]; t[n_] := Table[b[n] - b[n-i], {i, 1, n}]; Table[t[n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Alois P. Heinz *)

Extensions

Typo in definition (limits on i were wrong) corrected and reference added to Bohman's paper. N. J. A. Sloane, Feb 09 2012