# Definition

• $$\a(1) = 1\$$
• $$\a(2) = 2\$$
• $$\a(n)\$$ is smallest number $$\k>a(n-1)\$$ which avoids any 3-term arithmetic progression in $$\a(1), a(2), ..., a(n-1), k\$$.
• In other words, $$\a(n)\$$ is the smallest number $$\k>a(n-1)\$$ such that there does not exist $$\x, y\$$ where $$\0 and $$\a(y)-a(x) = k-a(y)\$$.

# Worked out example

For $$\n=5\$$:

We have $$\a(1), a(2), a(3), a(4) = 1, 2, 4, 5\$$

If $$\a(5)=6\$$, then $$\2, 4, 6\$$ form an arithmetic progression.

If $$\a(5)=7\$$, then $$\1, 4, 7\$$ form an arithmetic progression.

If $$\a(5)=8\$$, then $$\2, 5, 8\$$ form an arithmetic progression.

If $$\a(5)=9\$$, then $$\1, 5, 9\$$ form an arithmetic progression.

If $$\a(5)=10\$$, no arithmetic progression can be found.

Therefore $$\a(5)=10\$$.

Given $$\n\$$, output $$\a(n)\$$.

# Specs

• $$\n\$$ will be a positive integer.
• You can use 0-indexed instead of 1-indexed, in which case $$\n\$$ can be $$\0\$$. Please state it in your answer if you are using 0-indexed.

# Scoring

Since we are trying to avoid 3-term arithmetic progression, and 3 is a small number, your code should be as small (i.e. short) as possible, in terms of byte-count.

# Testcases

The testcases are 1-indexed. You can use 0-indexed, but please specify it in your answer if you do so.

1     1
2     2
3     4
4     5
5     10
6     11
7     13
8     14
9     28
10    29
11    31
12    32
13    37
14    38
15    40
16    41
17    82
18    83
19    85
20    86
10000 1679657


# References

• Related. (If I understand your challenge correctly.) – Martin Ender Aug 2 '16 at 14:56
• @MartinEnder You did understand my challenge correctly. – Leaky Nun Aug 2 '16 at 14:56

# Jelly, 4 bytes

Bḅ3‘


### How it works

This uses 0-based indexing and the primary definition from OEIS:

Szekeres's sequence: a(n)-1 in ternary = n-1 in binary

Bḅ3‘  Main link. Argument: n

B     Convert n to binary.
ḅ3   Convert from ternary to integer.
‘  Increment the result.


# Haskell, 37 36 32 bytes

Using the given formula in the OEIS entry, using 0-based indices. Thanks @nimi for 4 bytes!

a 0=1;a m=3*a(div m 2)-2+mod m 2


# Python 3, 28 bytes

lambda n:int(bin(n)[2:],3)+1


An anonymous function that takes input via argument and returns the result. This is zero-indexed.

How it works

lambda n    Anonymous function with input zero-indexed term index n
bin(n)      Convert n to a binary string..
...[2:]     ...remove 0b from beginning...
int(...,3)  ...convert from base-3 to decimal...
...+1       ...increment...
:...        and return


Try it on Ideone

# Python 3, 113 bytes

def f(n):
i=1;a=[]
for _ in range(n):
while any(i+x in[y*2for y in a]for x in a):i+=1
a+=[i]
return a[n-1]


Ideone it!

# Ruby, 28 24 bytes

Using the same method as Dennis, with 0-based indexes:

->n{n.to_s(2).to_i(3)+1}


Run the test cases on repl.it: https://repl.it/Cif8/1

## Pyke, 5 bytes

b2b3h


Try it here!

0-based indexing

Same formula as jelly answer

# APL (Dyalog Extended), 5 bytes

1+3⊥⊤


Try it online!

Jo King's suggestion.

# APL (Dyalog Unicode), 12 bytes

1+3⊥2(⊥⍣¯1)⊢


Try it online!

Based on Dennis' Jelly answer.

Outputs are zero indexed.

1+3⊥2(⊥⍣¯1)⊢
⊢ Take argument
2(⊥⍣¯1)  Encode in binary
3⊥         Decode from ternary

• With Dyalog Extended, this can be just 5 bytes! You can also save 2 bytes in Unicode – Jo King Sep 18 '20 at 13:23

# Java 7, 60 42 bytes

0 Indexed

int s(int n){return n<1?1:3*s(n/2)-2+n%2;}


Using the implicit sequence definition from OEIS. Thanks to Kevin for the improvement using only one return statement.

• Welcome to CGCC! :) You don't need the static , and you can also golf the if(n==0)return 1;return 3*s(n/2)-2+n%2; to return n<1?1:3*s(n/2)-2+n%s;. :) If you haven't seen it yet, tips for golfing in Java and tips for golfing in all languages might be interesting to read through. Enjoy your stay! – Kevin Cruijssen Sep 18 '20 at 13:54

## Java 8, 52 46 bytes

0 indexed.

i->Integer.valueOf(Integer.toString(i,2),3)+1;

• You don't need the return but you do need the semicolon afterwards – Leaky Nun Aug 3 '16 at 15:12
• This answer that say semicolons aren't counted; I could change it either way, but is counting semicolons the consensus? – Justin Aug 3 '16 at 15:14
• Eh, that's what they told me. I don't know if the consensus is as such. – Leaky Nun Aug 3 '16 at 15:24
• Alrighty, no return plus semi-colon is shorter than before anyway :) – Justin Aug 3 '16 at 15:30

# Pip, 11 bytes

1+(aTB2)FB3


Try it online!

Based on Dennis' Jelly answer. Zero Indexed.

# Japt, 4 bytes

0-indexed

Ò¢n3


Same as most other solutions:

Ò¢n3     :Implicit input of integer U
Ò        :Negation the bitwise NOT of
¢       :U converted to a binary string
n3     :Converted from a ternary string