A297464 -id:A297464 - cp's OEIS frontend

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

6, 16, 27, 36, 46, 57, 66, 75, 87, 96, 106, 117, 126, 136, 147, 156, 165, 177, 186, 196, 207, 216, 227, 237, 246, 255, 267, 276, 286, 297, 306, 316, 327, 336, 345, 357, 366, 376, 387, 396, 406, 417, 426, 435, 447, 456, 466, 477, 486, 497, 507, 516, 525, 537

Offset: 1

Peter Kagey, Dec 08 2015

nonn

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

Peter Kagey, Table of n, a(n) for n = 1..10000
Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, Using Walnut to solve problems from the OEIS, arXiv:2503.04122 [math.NT], 2025.
Robbert Fokkink and Gandhar Joshi, Anti-recurrence sequences, arXiv:2506.13337 [math.NT], 2025. See pp. 2, 11, 18.
William Lowell Putnam Competition, Problem B2, 2015.

S:= {$1..1000}: A:= NULL:
while nops(S) >= 3 do
  T:= S[1..3];
  s:= convert(T,`+`);
  S:= S[4..-1] minus {s};
  A:= A, s
od:
A; # Robert Israel, Dec 22 2015

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 *)
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]];
Take[A265389, 100]    (* Peter J. C. Moses, May 03 2018 *)

x = (1..10000).to_a
(0...1000).collect do
  y = x.shift(3).reduce(:+); x.delete_at x.index(y); y
end

A297465 Solution (b(n)) of the system of 4 complementary equations in Comments.

Original entry on oeis.org

2, 5, 9, 12, 15, 19, 22, 25, 29, 32, 35, 39, 42, 45, 49, 52, 55, 59, 62, 65, 69, 72, 76, 79, 82, 85, 89, 92, 95, 99, 102, 105, 109, 112, 115, 119, 122, 125, 129, 132, 135, 139, 142, 145, 149, 152, 155, 159, 162, 166, 169, 172, 175, 179, 182, 185, 189, 192

Offset: 0

Clark Kimberling, Apr 22 2018

Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3;:
a(n) = least new;
b(n) = least new;
c(n) = least new;
d(n) = a(n) + b(n) + c(n);
where "least new k" means the least positive integer not yet placed.
***
Conjecture: for all n >= 0,
0 <= 10n - 6 - 3 a(n) <= 2
0 <= 10n - 2 - 3 b(n) <= 3
0 <= 10n +1 - 3 c(n) <= 3
0 <= 10n - 3 - d(n) <= 2
***
The sequences a,b,c,d partition the positive integers.  The sequence d can be called the "anti-tribonacci sequence"; viz., if sequences a and b are defined as above, and c(n) is defined by c(n) = a(n) + b(n), then the resulting system of 3 complementary sequences gives c = A036554, the "anti-Fibonacci sequence."

			n:   0    1    2    3    4    5    6    7    8    9
a:   1    4    8   11   14   18   21   24   28   31
b:   2    5    9   12   15   19   22   25   29   32
c:   3    7   10   13   17   20   23   26   30   33
d:   6   16   27   36   46   57   66   75   87   96

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A036554, A299634, A297464, A297466, A265389.

z = 400;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3}; d = {}; AppendTo[d, Last[a] + Last[b] + Last[c]];
Do[{AppendTo[a, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[b, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[c, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[d, Last[a] + Last[b] + Last[c]]}, {z}];
Take[a, 100]  (* A297464 *)
Take[b, 100]  (* A297465 *)
Take[c, 100]  (* A297466 *)
Take[d, 100]  (* A265389 *)

A297466 Solution (c(n)) of the system of 4 complementary equations in Comments.

Original entry on oeis.org

3, 7, 10, 13, 17, 20, 23, 26, 30, 33, 37, 40, 43, 47, 50, 53, 56, 60, 63, 67, 70, 73, 77, 80, 83, 86, 90, 93, 97, 100, 103, 107, 110, 113, 116, 120, 123, 127, 130, 133, 137, 140, 143, 146, 150, 153, 157, 160, 163, 167, 170, 173, 176, 180, 183, 187, 190, 193

Offset: 0

Clark Kimberling, Apr 22 2018

Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3:
a(n) = least new;
b(n) = least new;
c(n) = least new;
d(n) = a(n) + b(n) + c(n);
where "least new k" means the least positive integer not yet placed.
***
Conjecture: for all n >= 0,
0 <= 10n - 6 - 3 a(n) <= 2
0 <= 10n - 2 - 3 b(n) <= 3
0 <= 10n + 1 - 3 c(n) <= 3
0 <= 10n - 3 - d(n) <= 2
***
The sequences a,b,c,d partition the positive integers.  The sequence d can be called the "anti-tribonacci sequence"; viz., if sequences a and b are defined as above, and c(n) is defined by c(n) = a(n) + b(n), then the resulting system of 3 complementary sequences gives c = A036554, the "anti-Fibonacci sequence."
The linear recurrence with signature (1,0,0,0,0,0,0,0,1,-1) and g.f. (3 + 4*x + 3*x^2 + 3*x^3 + 4*x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8)/(1 - x - x^9 + x^10) first differs from this sequence at a(67). - Georg Fischer, Jul 17 2025

			n:   0    1    2    3    4    5    6    7    8    9
a:   1    4    8   11   14   18   21   24   28   31
b:   2    5    9   12   15   19   22   25   29   32
c:   3    7   10   13   17   20   23   26   30   33
d:   6   16   27   36   46   57   66   75   87   96

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A036554, A299634, A297464, A297465, A265389.

z = 400;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3}; d = {}; AppendTo[d, Last[a] + Last[b] + Last[c]];
Do[{AppendTo[a, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[b, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[c, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[d, Last[a] + Last[b] + Last[c]]}, {z}];
Take[a, 100]  (* A297464 *)
Take[b, 100]  (* A297465 *)
Take[c, 100]  (* A297466 *)
Take[d, 100]  (* A265389 *)

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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.

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A297465 Solution (b(n)) of the system of 4 complementary equations in Comments.

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A297466 Solution (c(n)) of the system of 4 complementary equations in Comments.

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