# Longest arithmetic subsequence

Given a non empty finite sequence of integers, return an arithmetic subsequence of maximal length.

If there are multiple of the same maximal length, any of them can be returned.

### Definitions:

An arithmetic sequence is a sequence $$\a(1),a(2),a(3),a(4),...\$$ such that there is a constant $$\c\$$ such that $$\a(m+1)-a(m) = c\$$ for all $$\m\$$. In other words: The difference between two subsequent terms is constant.

Given a sequence $$\b(1),b(2),b(3),b(4),...\$$ a subsequence is a sequence $$\b(s(1)),b(s(2)),b(s(3)),b(s(4)),...\$$ where $$\1 <= s(1)\$$ and $$\s(m) < s(m+1)\$$ for all $$\m\$$. In other words: Take the original sequence and remove as many entries as you want.

### Examples

Input                     Output
[4,1,2,3,6,5]             [1,3,5] or [1,2,3]
[5,4,2,-1,-2,-4,-4]       [5,2,-1,-4]
[1,2,1,3,1,4,1,5,1]       [1,1,1,1,1] or [1,2,3,4,5]
[1]                       [1]


Some longer test cases:

Length: 25
Input: [-9,0,5,15,-1,4,17,-3,20,13,15,9,0,-6,11,17,17,9,26,11,5,11,3,16,25]
Output: [15,13,11,9] or [17,13,9,5]

Length: 50
Input: [35,7,37,6,6,33,17,33,38,30,38,12,37,49,44,5,19,19,35,30,40,19,11,5,39,11,20,28,12,33,25,8,40,6,15,12,27,5,21,6,6,40,15,31,49,22,35,38,22,33]
Output: [6,6,6,6,6] or [39,33,27,21,15]

Length: 100
Input: [6,69,5,8,53,10,82,82,73,15,66,52,98,65,81,46,44,83,9,14,18,40,84,81,7,40,53,42,66,63,30,44,2,99,17,11,38,20,49,34,96,93,6,74,27,43,55,95,42,99,31,71,67,54,70,67,18,13,100,18,4,57,89,67,20,37,47,99,16,86,65,38,20,43,49,13,59,23,39,59,26,30,62,27,83,99,74,35,59,11,91,88,82,27,60,3,43,32,17,18]
Output: [6,18,30,42,54] or [8,14,20,26,32] or [46,42,38,34,30] or [83,63,43,23,3] or [14,17,20,23,26] or [7,17,27,37,47] or [71,54,37,20,3]


### Background

I got this idea when I recalled the Green-Tao-Theorem from 2004, which states that the sequence of primes contains finite arithmetic sequences of arbitrary length.

# Jelly, 8 bytes

ŒPIE$ÐfṪ  ### How it works ŒPIE$ÐfṪ  Main link. Argument: A (list of integers)

ŒP        Powerset; generate all sublists of A, sorted by length.
Ðf   Filter the powerset by the link to the left:
\$       Combine the two atoms to the left into a monadic link.
I           Compute all increments.
E          Test whether they're all equal.
This returns all arithmetic subsequences, sorted by length.
Ṫ  Tail; extract the last sequence.


## Pyth, 12 11 bytes

ef!P{-VTtTy


Test suite.

          y  powerset of implicit input, generate all subsequences
ef       T   find the last subsequence (sorted increasing in length) where...
Tt      bifurcate over tail, giving [1,2,3,4,5] [2,3,4,5]
-V        vectorize over -, giving differences of each consecutive pair
{          dedup (remove duplicate elements)
P           pop, resulting in [] if all differences were equal
!            boolean not, resulting in True if all differences were equal


Thanks to @LeakyNun for a byte!

# MATL, 19181716 18 bytes

1 byte saved (and 2 bytes added back) thanks to Luis!

"GX@XN!"@dun2<?vx@


A fairly naive approach which brute force checks all ordered permutations of the input. Obviously this can take a while for longer sequences. To save a byte, I have started with the smallest sub-sequences (length = 1) and worked up to the larger sequences (length = N).

Try it Online!

Explanation

                % Impilicitly grab input array (N)
"               % For each value in this array
G           % Explicitly grab the input
X@          % Loop index, will be [1, 2, ... length(N)] as we iterate
XN          % Determine all permutations of k elements (nchoosek). The output is
% each k-length permutation of the input as a different row. Order doesn't
% matter so the result is always ordered the same as the input
!           % Take the transpose so each k-length permutation is a column
"           % For each column
d       % Compute the difference between successive elements
un      % Determine the number of unique differences
2<?     % If there are fewer than 2 unique values
vx  % Vertically concatenate everything on the stack so far and delete it
@   % Push the current permuatation to the stack
% Implicit end of if statement
% Implicit end of for loop
% Implicit end of for loop
% Implicitly display the stack

• @LuisMendo Thanks! I always wondered how to get the loop iteration #. – Suever May 29 '16 at 3:45
• @LuisMendo Oh good catch, you're right. That double diff gives an empty array which can't be negated. – Suever May 29 '16 at 4:01

# Husk, 6 bytes

►oEẊ-Ṗ


Try it online!

## Explanation

The return value of E is useful here for getting the longest list that satisfies a condition.

►oEẊ-Ṗ   Implicit argument: a list.
Ṗ   List of all its subsequences.
►        Find one that maximizes:
o        Composition of two functions:
Ẋ       1. Apply to adjacent elements:
-         Subtraction.
E        2. Are all elements equal?
Returns length+1 if they are, 0 if not.


# Python 2, 12411598 97 bytes

p=[[]]
for n in input():p+=[x+[n][:2>len(x)or n-x[-1]==x[1]-x[0]]for x in p]
print max(p,key=len)


Very slow and memory intensive. Test it on Ideone.

### Alternate version, 98 bytes

p={()}
for n in input():p|={x+(n,)[:2>len(x)or n-x[-1]==x[1]-x[0]]for x in p}
print max(p,key=len)


This completes all test cases instantly. Test it on Ideone.

• byte or speed, that is the question – downrep_nation May 29 '16 at 9:06

# Pyth checkout 8593c76, March 24th, 10 bytes

efq-VTtT)y


This is exactly the same as Doorknob's answer, except that back in march, there was a 2 byte function (q ... )) which checked whether all of the elements of a list were the same, which is the same as !P{, which is the best you can do currently.

## JavaScript (ES6), 157 bytes

a=>{for(m=i=0,l=a.length;i<l;i++)for(j=i;++j<l;)for(t=n=a[k=i],c=0;k<l;k++)a[k]==t&&(t+=a[j]-n,++c>m)?q=a[m=c,p=n,j]-n:q;return a.slice(-m).map(_=>(p+=q)-q)}


Almost 20 times longer than the Jelly answer... Ungolfed:

function subsequence(a) {
var max = 0;
for (var i = 0; i < a.length; i++) {
for (var j = i + 1; j < a.length; j++) {
var target = a[i];
var count = 0;
for (var k = i; k < a.length; k++) {
if (a[k] == target) {
count++;
target += a[j] - a[i];
if (count > max) {
max = count;
start = a[i];
step = a[j] - a[i];
}
}
}
}
}
var result = new Array(max);
for (var i = 0; i < max; i++) {
result[i] = start + step * i;
}
return result;
}


# Japt-h, 13 bytes

à ñÊkÈän än d


Try it