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

1,3

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.

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.

Example

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}

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 *)

Crossrefs

Cf. A005318.

Sequence in context: A132993 A106408 A143061 * A037254 A316939 A259478

Adjacent sequences: A096855 A096856 A096857 * A096859 A096860 A096861

Keyword

nonn,tabl

Author

N. J. A. Sloane, Aug 18 2004

Extensions

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

Status

approved