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A265389 The sums from the following procedure: from the list of positive integers, repeatedly remove the first three numbers and their sum.

%I A265389 #39 Jun 23 2025 10:54:20
%S A265389 6,16,27,36,46,57,66,75,87,96,106,117,126,136,147,156,165,177,186,196,
%T A265389 207,216,227,237,246,255,267,276,286,297,306,316,327,336,345,357,366,
%U A265389 376,387,396,406,417,426,435,447,456,466,477,486,497,507,516,525,537
%N A265389 The sums from the following procedure: from the list of positive integers, repeatedly remove the first three numbers and their sum.
%C A265389 This sequence is a solution, along with three other sequences, of a system of four complementary equations; see A297464.  It is the "anti-tribonacci" sequence, in analogy with the anti-Fibonacci sequence, A075326. - _Clark Kimberling_, Apr 22 2018
%H A265389 Peter Kagey, <a href="/A265389/b265389.txt">Table of n, a(n) for n = 1..10000</a>
%H A265389 Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, <a href="https://arxiv.org/abs/2503.04122">Using Walnut to solve problems from the OEIS</a>, arXiv:2503.04122 [math.NT], 2025.
%H A265389 Robbert Fokkink and Gandhar Joshi, <a href="https://arxiv.org/abs/2506.13337">Anti-recurrence sequences</a>, arXiv:2506.13337 [math.NT], 2025. See pp. 2, 11, 18.
%H A265389 William Lowell Putnam Competition, <a href="http://kskedlaya.org/putnam-archive/2015.pdf">Problem B2</a>, 2015.
%p A265389 S:= {$1..1000}: A:= NULL:
%p A265389 while nops(S) >= 3 do
%p A265389   T:= S[1..3];
%p A265389   s:= convert(T,`+`);
%p A265389   S:= S[4..-1] minus {s};
%p A265389   A:= A, s
%p A265389 od:
%p A265389 A; # _Robert Israel_, Dec 22 2015
%t A265389 f[n_] := Block[{a = {}, r = Range@ n, s}, Do[If[Length@ r > 4, s = Total@ Take[r, 3 ]; AppendTo[a, s]; r = Drop[#, 3] &@ DeleteCases[r, x_ /; x == s], Break[]], {k, n}]; a]; f@ 184 (* _Michael De Vlieger_, Dec 22 2015 *)
%t A265389 morph = Nest[Flatten[# /. {0 -> {1, 2, 0}, 1 -> {1, 1, 0}, 2 -> {1, 0, 0}}] &, {0}, 9]; A265389 = Accumulate[Prepend[Drop[Flatten[morph /. Thread[{0, 1, 2} -> {{1, 1, 4}, {1, 2, 3}, {1, 3, 2}}]], 1] + 8, 6]];
%t A265389 Take[A265389, 100]    (* _Peter J. C. Moses_, May 03 2018 *)
%o A265389 (Ruby)
%o A265389 x = (1..10000).to_a
%o A265389 (0...1000).collect do
%o A265389   y = x.shift(3).reduce(:+); x.delete_at x.index(y); y
%o A265389 end
%K A265389 nonn
%O A265389 1,1
%A A265389 _Peter Kagey_, Dec 08 2015