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
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