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%I #29 May 24 2021 00:09:01
%S 1,-2,4,-3,-5,11,-6,-7,8,9,10,12,13,-14,-16,-15,-17,18,19,-20,21,22,
%T 23,24,25,-26,-27,-34,-28,-29,-30,31,32,33,35,-36,37,38,39,40,41,-42,
%U -43,-44,-57,-45,-46,-47,48,49,-50,51,52,53,-55,54,56,58,59,60,61,-62,-63,-64,-65,-83,-66,-67,-68,-69,70,71
%N Successive sums of successive terms produce the successive natural numbers (see the Comments section).
%C The 1st term sums up to 1;
%C the next 2 terms sum up to 2;
%C the next 3 terms sum up to 3;
%C the next 4 terms sum up to 4;
%C ... the next k terms sum up to k.
%H Carole Dubois, <a href="/A338161/b338161.txt">Table of n, a(n) for n = 1..406</a>
%H Carole Dubois, <a href="/A338161/a338161_1.txt">Program (Python)</a>
%e 1 = 1 (1 term);
%e 2 = - 2 + 4 (2 terms);
%e 3 = - 3 - 5 + 11 (3 terms);
%e 4 = - 6 - 7 + 8 + 9 (4 terms);
%e 5 = 10 + 12 + 13 - 14 - 16 (5 terms);
%e 6 = - 15 - 17 + 18 + 19 - 20 + 21 (6 terms); etc.
%e How are the plus and minus signs split between the terms to get the above six equations? Here is the method -- with an example:
%e 1) no absolute value of any term can be present twice or more in the sequence;
%e 2) to start a new equation, always use the set of smallest absolute values not yet used; say, for the above 5-term equation, that they are a, b, c, d and e;
%e 3) the set of unused values for a, b, c, d and e is here 10, 12, 13, 14, 15;
%e 4) try all the possible mix of values and signs to find one or more solutions (the try 5 = 10 + 12 - 13 - 14 + 15, for instance, doesn't work as we would get 5 = 10);
%e 5) if no such mix leads to a solution (which is the case here), add 1 to the biggest integer of the values' set and try again;
%e 6) the above set would then become 10, 12, 13, 14, 16 -- and a quick computer search gives the solution 5 = 10 + 12 + 13 - 14 - 16;
%e 7) had we found more than one solution, we would have kept the lexicographically earliest one (-10 comes before +10);
%e 8) if a new mix doesn't lead to a solution, add again 1 to the biggest integer of the values' set and try again; etc.
%o (Python) # see Link section
%Y Cf. A330903.
%K sign
%O 1,2
%A _Eric Angelini_ and _Carole Dubois_, Oct 14 2020
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