%I #25 Aug 20 2024 22:37:29

%S 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,

%T 44,40,60,71,77,80,82,83,84,77,117,137,148,154,157,159,160,161,148,

%U 225,265,285,296,302,305,307,308,309,285,433,510,550,570,581,587,590,592,593,594

%N 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().

%C 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

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

%D J. H. Conway and R. K. Guy, Solution of a problem of Erdős, Colloq. Math. 20 (1969), p. 307.

%D 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.

%D R. K. Guy, Unsolved Problems in Number Theory, C8.

%H Alois P. Heinz, <a href="/A096858/b096858.txt">Rows n = 1..141, flattened</a>

%H Tom Bohman, <a href="http://dx.doi.org/10.1090/S0002-9939-96-03653-2">A sum packing problem of Erdős and the Conway-Guy sequence</a>, Proc. AMS 124, (No. 12, 1996), pp. 3627-3636.

%e The triangle begins:

%e {1}

%e {1,2}

%e {2,3,4}

%e {3,5,6,7}

%e {6,9,11,12,13}

%e {11,17,20,22,23,24}

%e {20,31,37,40,42,43,44}

%e {40,60,71,77,80,82,83,84}

%e {77,117,137,148,154,157,159,160,161}

%e {148,225,265,285,296,302,305,307,308,309}

%e {285,433,510,550,570,581,587,590,592,593,594}

%e {570,855,1003,1080,1120,1140,1151,1157,1160,1162,1163,1164}

%e {1120,1690,1975,2123,2200,2240,2260,2271,2277,2280,2282,2283,2284}

%e {2200,3320,3890,4175,4323,4400,4440,4460,4471,4477,4480,4482,4483,4484}

%e {4323,6523,7643,8213,8498,8646,8723,8763,8783,8794,8800,8803,8805,8806,8807}

%p b:= proc(n) option remember;

%p `if`(n<2, n, 2*b(n-1) -b(n-1-floor(1/2 +sqrt(2*n-2))))

%p end:

%p T:= n-> seq(b(n)-b(n-i), i=1..n):

%p seq(T(n), n=1..15); # _Alois P. Heinz_, Nov 29 2011

%t 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_ *)

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

%K nonn,tabl

%O 1,3

%A _N. J. A. Sloane_, Aug 18 2004

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