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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().

%I A096858 #25 Aug 20 2024 22:37:29
%S A096858 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 A096858 44,40,60,71,77,80,82,83,84,77,117,137,148,154,157,159,160,161,148,
%U A096858 225,265,285,296,302,305,307,308,309,285,433,510,550,570,581,587,590,592,593,594
%N 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().
%C A096858 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 A096858 It is also conjectured that in some sense this triangle is optimal. See A005318 for further information and additional references.
%D A096858 J. H. Conway and R. K. Guy, Solution of a problem of Erdős, Colloq. Math. 20 (1969), p. 307.
%D A096858 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 A096858 R. K. Guy, Unsolved Problems in Number Theory, C8.
%H A096858 Alois P. Heinz, <a href="/A096858/b096858.txt">Rows n = 1..141, flattened</a>
%H A096858 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 A096858 The triangle begins:
%e A096858 {1}
%e A096858 {1,2}
%e A096858 {2,3,4}
%e A096858 {3,5,6,7}
%e A096858 {6,9,11,12,13}
%e A096858 {11,17,20,22,23,24}
%e A096858 {20,31,37,40,42,43,44}
%e A096858 {40,60,71,77,80,82,83,84}
%e A096858 {77,117,137,148,154,157,159,160,161}
%e A096858 {148,225,265,285,296,302,305,307,308,309}
%e A096858 {285,433,510,550,570,581,587,590,592,593,594}
%e A096858 {570,855,1003,1080,1120,1140,1151,1157,1160,1162,1163,1164}
%e A096858 {1120,1690,1975,2123,2200,2240,2260,2271,2277,2280,2282,2283,2284}
%e A096858 {2200,3320,3890,4175,4323,4400,4440,4460,4471,4477,4480,4482,4483,4484}
%e A096858 {4323,6523,7643,8213,8498,8646,8723,8763,8783,8794,8800,8803,8805,8806,8807}
%p A096858 b:= proc(n) option remember;
%p A096858       `if`(n<2, n, 2*b(n-1) -b(n-1-floor(1/2 +sqrt(2*n-2))))
%p A096858     end:
%p A096858 T:= n-> seq(b(n)-b(n-i), i=1..n):
%p A096858 seq(T(n), n=1..15);  # _Alois P. Heinz_, Nov 29 2011
%t A096858 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 A096858 Cf. A005318, A005230 (column 1 of triangle).
%K A096858 nonn,tabl
%O A096858 1,3
%A A096858 _N. J. A. Sloane_, Aug 18 2004
%E A096858 Typo in definition (limits on i were wrong) corrected and reference added to Bohman's paper. _N. J. A. Sloane_, Feb 09 2012