

A096858


Triangle read by rows in which row n gives the nset obtained as the differences {b(n)b(ni), 0 <= i <= n1}, where b() = A005318().


7



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
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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. 141154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. NorthHolland 1982.
R. K. Guy, Unsolved Problems in Number Theory, C8.


LINKS



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(n1) b(n1floor(1/2 +sqrt(2*n2))))
end:
T:= n> seq(b(n)b(ni), i=1..n):


MATHEMATICA

b[n_] := b[n] = If[n < 2, n, 2*b[n1]  b[n1Floor[1/2 + Sqrt[2*n2]]]]; t[n_] := Table[b[n]  b[ni], {i, 1, n}]; Table[t[n], {n, 1, 15}] // Flatten (* JeanFrançois Alcover, Jan 14 2014, after Alois P. Heinz *)


CROSSREFS



KEYWORD



AUTHOR



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



