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Lexicographically earliest sequence of positive integers such that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression in any order.
+10
8
1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 9, 4, 2, 5, 11, 2, 2, 4, 1, 1, 5, 1, 1, 10, 2, 2, 4, 1, 1, 4, 4, 10, 10, 4, 8, 10, 10, 2, 4, 1, 2, 5, 4, 10, 10, 4, 2, 8, 8, 5, 8, 5, 13, 13, 17, 5, 13, 2, 11, 17, 10, 10, 13, 13
COMMENTS
This sequence avoids all six of the six permutations of a set of three integers in arithmetic progression. For example, the set {1,2,3} can be ordered as tuples (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1).
This sequence is part of a family of variants avoiding different permutations of arithmetic progressions at indices in arithmetic progression:
- A100480 (offset 1), A006997 (offset 0): Prohibits 1,1,1 and progressions of common difference 0. - A309890: Prohibits 1,2,3 or progressions of the form c, c+d, c+2d, for all d >= 0. - A373111: Prohibits 1,3,2 or progressions of the form c, c+2d, c+d, for all d >= 0. - A371457: Prohibits 2,1,3 or progressions of the form c, c-d, c+d, for all d >= 0. - A371632: Prohibits 2,3,1 or progressions of the form c, c+d, c-d, for all d >= 0. - A373010: Prohibits 3,1,2 or progressions of the form c, c-2d, c-d, for all d>=0. - A373052: Prohibits 3,2,1 or progressions of the form c, c-d, c-2d, for all d>=0. With the sequences prohibiting the six permutations above, there are a total of 64 sequences which prohibit some combination of these six permutations of an arithmetic progression. At least two more of these are in the OEIS:
- A229037 ("forest fire sequence"): Prohibits (progressions of the same general form as) 1,2,3 and 3,2,1 . - A361933 (the present sequence): Prohibits all six permutations.
EXAMPLE
a(28) cannot be 1 because then a(26)=5, a(27)=9, and a(28)=1 could be rearranged to form an arithmetic progression (1, 5, 9). The numbers 2-8 could also create an arithmetic progression so a(28)=9.
PROG
(PARI) \\ See Links section.
Lexicographically earliest sequence of positive integers such that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form a progression of the form p, p-2*q, p-q, where q >= 0.
+10
6
1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 3, 1, 1, 2, 4, 3, 3, 4, 3, 2, 2, 4, 4, 2, 2, 5, 1, 1, 3, 1, 1, 2, 4, 4, 5, 1, 1, 3, 1, 1, 5, 4, 5, 3, 6, 5, 6, 5, 4, 6, 6, 4, 3, 4, 3, 3, 4, 3, 6, 2, 6, 5, 7, 3, 6, 6, 3, 2, 7, 6, 7, 5, 5, 2, 2, 6, 2, 2, 4, 5, 1, 1, 2, 1, 1, 5, 2, 6, 7
COMMENTS
This sequence avoids one of the six permutations of a set of three integers in arithmetic progression. For example, the set {1,2,3} can be ordered as tuples (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). In this sequence, we avoid (3,1,2) and other progressions of the form p, p-2*q, p-q, for all q >= 0.
Lexicographically earliest sequence of positive integers such that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form a weakly decreasing arithmetic progression.
+10
6
1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 5, 4, 4, 5, 1, 3, 2, 4, 1, 1, 2, 1, 3, 2, 4, 2, 5, 1, 2, 2, 1, 3, 3, 4, 3, 4, 5, 2, 4, 3, 5, 5, 6, 3, 4, 3, 6, 4, 4, 5, 5, 4, 5, 5, 6, 6, 7, 6, 6, 7, 7, 5, 8, 6, 8, 6, 7, 7, 2, 7, 7, 2, 8
PROG
(PARI) \\ See Links section.
Lexicographically earliest sequence of positive integers such that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form a progression of the form p, p+q, p-q, where q >= 0.
+10
5
1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 2, 4, 1, 3, 2, 1, 3, 2, 3, 4, 1, 2, 3, 4, 3, 4, 4, 5, 4, 1, 5, 5, 4, 1, 4, 2, 5, 5, 6, 5, 5, 6, 6, 7, 7, 3, 7, 6, 6, 8, 6, 6, 5, 7, 7, 8, 7, 1, 8, 8, 9, 9, 8, 5, 3, 9, 9, 10, 9, 6, 8, 8, 5, 9, 9, 5, 8, 6, 10, 1, 7, 10, 6, 6, 4, 4, 8, 3, 10
COMMENTS
This sequence avoids one of the six permutations of a set of three integers in arithmetic progression. For example, the set {1,2,3} can be ordered as tuples (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). In this sequence, we avoid (2,3,1) and other progressions of the form p, p+q, p-q, for all q >= 0.
Lexicographically earliest sequence of positive integers such that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form a progression of the form c, c+2d, c+d, where d >= 0.
+10
5
1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 3, 1, 1, 4, 4, 3, 2, 3, 3, 4, 4, 5, 4, 4, 5, 5, 1, 1, 4, 1, 1, 5, 5, 6, 5, 1, 1, 6, 1, 1, 2, 2, 5, 2, 2, 5, 6, 6, 7, 2, 2, 7, 2, 2, 8, 7, 6, 5, 6, 8, 8, 9, 5, 6, 9, 8, 9, 2, 2, 7, 3, 2, 8, 2, 3, 8, 7, 3, 7, 4, 1, 1, 6, 1, 1, 9, 8
COMMENTS
This sequence avoids one of the six permutations of a set of three integers in arithmetic progression. For example, the set {1,2,3} can be ordered as tuples (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). In this sequence, we avoid (1,3,2) and other progressions of the form c, c+2d, c+d, for all d >= 0.
Array based on the Stanley sequence S(0), A005836, by antidiagonals.
+10
3
0, 2, 1, 7, 5, 3, 21, 8, 6, 4, 23, 22, 16, 11, 9, 64, 26, 24, 17, 14, 10, 69, 65, 50, 25, 19, 15, 12, 71, 70, 67, 53, 48, 20, 18, 13, 193, 80, 78, 68, 59, 49, 34, 29, 27, 207, 194, 152, 79, 73, 62, 51, 35, 32, 28, 209, 208, 196, 161, 150, 74, 63, 52, 43, 33, 30
COMMENTS
This array is similar to a dispersion in that the first column is the minimal nonnegative sequence that contains no 3-term arithmetic progression, and each next column is the minimal sequence consisting of the numbers rejected from the previous column that contains no 3-term arithmetic progression.
A100480(n) describes which column n is sorted into.
The columns of the array form the greedy partition of the nonnegative integers into sequences that contain no 3-term arithmetic progression. - Robert Israel, Feb 03 2016
EXAMPLE
From the top-left corner, this array starts:
0 2 7 21 23 64
1 5 8 22 26 65
3 6 16 24 50 67
4 11 17 25 53 68
9 14 19 48 59 73
10 15 20 49 62 74
MAPLE
M:= 20: # to get the first M antidiagonals
for i from 1 to M do B[i]:= {}: F[i]:= {}: od:
countdowns:= Vector(M, j->M+1-j):
for x from 0 while max(countdowns) > 0 do
for i from 1 do
if not member(x, F[i]) then
F[i]:= F[i] union map(y -> 2*x-y, B[i]);
B[i]:= B[i] union {x};
countdowns[i]:= countdowns[i] - 1;
break
fi
od;
od:
seq(seq(B[n+1-i][i], i=1..n), n=1..M); # Robert Israel, Feb 03 2016
PROG
(MATLAB)
% to get first M antidiagonals using x up to N
B = cell(1, M);
F = zeros(M, N+1);
countdowns = [M:-1:1];
for x=0:N
if max(countdowns) == 0
break
end
for i=1:M
if F(i, x+1) == 0
newforb = 2*x - B{i};
newforb = newforb(newforb <= N & newforb >= 1);
F(i, newforb+1) = 1;
B{i}(end+1) = x;
countdowns(i) = countdowns(i)-1;
break
end
end
end
if max(countdowns) > 0
[~, jmax] = max(countdowns);
jmax = jmax(1);
error ('Need larger N: B{%d} has only %d elements', jmax, numel(B{jmax}));
end
A = zeros(1, M*(M+1)/2);
k = 0;
for n=1:M
for i=1:n
k=k+1;
A(k) = B{n+1-i}(i);
end
end
Lexicographically earliest sequence of positive integers such that no three equal terms appear at distinct indices that are the side lengths of a triangle.
+10
3
1, 1, 1, 2, 1, 2, 3, 1, 3, 2, 4, 4, 1, 5, 5, 2, 3, 6, 6, 7, 1, 7, 4, 8, 8, 2, 3, 9, 5, 9, 10, 10, 11, 1, 4, 11, 6, 12, 12, 13, 13, 2, 7, 3, 5, 14, 14, 15, 8, 15, 16, 16, 17, 17, 1, 6, 18, 4, 9, 18, 19, 19, 10, 20, 7, 20, 21, 2, 11, 21, 3, 22, 22, 5, 8, 23, 12, 23, 24, 24, 13, 25, 25, 26, 26, 27, 27, 28, 1, 9, 28, 29, 4
COMMENTS
In a triangle, the sum of any two side lengths is greater than that of the third, so that x + y > z.
So if x < y and a(x) = a(y) = t then we cannot have a(z) = t for any z in the range y < z < x+y.
Another way to construct the sequence: Place 1's at the earliest permitted positions (in this case, at Fibonacci indices). Each subsequent value (2’s, 3’s, etc.) is placed at the earliest permitted indices not already occupied by a smaller value. For example, 3's could be placed in a Fibonacci pattern beginning with 7, 9 (7, 9, 16, 25, etc.), but i=7+9=16 is already occupied by the value 2, so 3 gets the next smallest position i=17. i=9+17=26 is again occupied by a 2, so we give 3 the next smallest unoccupied position i=27.
MATHEMATICA
list={1}; Do[k=1; While[lst=Join[list, {k}]; !And@@(And@@(({a, b, c}=#; (-a+b+c)(a-b+c)(a+b-c))<=0&/@Subsets[Flatten[Position[lst, #]], {3}])&/@Union@lst), k++]; AppendTo[list, k], {n, 92}]; list (* Giorgos Kalogeropoulos, Feb 20 2024 *)
PROG
(Python)
from itertools import combinations as C, count, islice
def agen(): # generator of terms
yield from [1, 1, 1]
sides = {1: [1, 2, 3]}
for n in count(4):
an = next(an for an in count(1) if an not in sides or all(not all((n<b+c, b<n+c, c<n+b)) for b, c in C(sides[an], 2)))
yield an
if an not in sides: sides[an] = []
sides[an].append(n)
Lexicographically earliest sequence of positive integers such that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form a progression of the form p, p-q, p+q, where q >= 0.
+10
3
1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 4, 3, 2, 6, 5, 5, 6, 3, 4, 3, 4, 1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 6, 4, 2, 4, 6, 8, 6, 5, 8, 4, 6, 2, 7, 5, 11, 5, 5, 7, 6, 11, 4, 9, 6, 7, 9, 7, 5, 4, 3, 8, 9, 5, 5, 8, 3, 5, 3, 3, 1, 1, 2, 1, 1, 2
COMMENTS
This sequence avoids one of the six permutations of a set of three integers in arithmetic progression. For example, the set {1,2,3} can be ordered as tuples (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). In this sequence, we avoid (2,1,3) and other progressions of the form p, p-q, p+q, for all q >= 0.
Lexicographically earliest sequence of positive integers such that all equal terms appear at mutually coprime indices.
+10
1
1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 4, 8, 1, 9, 1, 10, 5, 11, 1, 12, 2, 13, 7, 14, 1, 15, 1, 16, 8, 17, 3, 18, 1, 19, 10, 20, 1, 21, 1, 22, 11, 23, 1, 24, 2, 25, 13, 26, 1, 27, 6, 28, 14, 29, 1, 30, 1, 31, 16, 32, 7, 33, 1, 34, 17, 35, 1, 36, 1, 37, 19
COMMENTS
See A279119 for the same sequence with numbers including 0. See A055396 for a similar sequence where all equal terms share a factor > 1.
EXAMPLE
a(4)=2 because if we had a(4)=1, then i=2 and i=4, which are not coprime indices, would have the same value 1. So a(4)=2, which is a first occurrence.
a(9)=2 because if we had a(9)=1, i=3 and i=9, would have the same value despite not being coprime indices. a(9) can be 2 because the only other index with a 2 is a(4)=2 and 4 is coprime to 9.
a(15)=4 because 4 is the smallest value such that every previous index at which a 4 occurs is coprime to i=15. In this case, 4 has only occurred at i=8 and 8 is coprime to 15.
PROG
(Python)
from math import gcd, lcm
from itertools import combinations as C, count, islice
def agen(): # generator of terms
yield from [1, 1, 1]
lcms = {1: 6}
for n in count(4):
an = next(an for an in count(1) if an not in lcms or gcd(lcms[an], n) == 1)
yield an
if an not in lcms: lcms[an] = n
else: lcms[an] = lcm(lcms[an], n)
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