Generate Linus Sequence

Question

Definition

From the description on OEIS A006345:

To find a(n), consider either a 1 or a 2. For each, find the longest repeated suffix, that is, for each of a(n)=1,2, find the longest sequence s with the property that the sequence a(1),...,a(n) ends with ss. Use the digit that results in the shorter such suffix. a(1) = 1.

Worked-out Example

a(1)=1.

If a(2)=1, we will have the sequence 1 1 where the longest doubled substring from the end is 1. If a(2)=2 instead, then it would be the empty substring. Therefore a(2)=2.

When n=6, we choose between 1 2 1 1 2 1 and 1 2 1 1 2 2. In the first choice, 1 2 1 is doubled consecutively from the end. In the second choice, it is 2 instead. Therefore, a(6)=2.

When n=9, we choose between 1 2 1 1 2 2 1 2 1 and 1 2 1 1 2 2 1 2 2. In the first choice, the longest doubled consecutive substring is 2 1, while in the second choice 1 2 2 is doubled consecutively at the end. Therefore a(9)=1.

Task

Given n, return a(n).

Specs

• n will be positive.
• You can use 0-indexed instead of 1-indexed. In that case, please state so in your answer. Also, in that case, n can be 0 also.

Testcases

The testcases are 1-indexed. However, you can use 0-indexed.

n  a(n)
1  1
2  2
3  1
4  1
5  2
6  2
7  1
8  2
9  1
10 1
11 2
12 1
13 2
14 2
15 1
16 1
17 2
18 1
19 1
20 1

References

• WolframMathWorld
• Obligatory OEIS A006345

Answer 1

Haskell, 146 140 137 133 118 bytes

s!l|take l s==take l(drop l s)=l|1<2=s!(l-1)
g[w,x]|w<x=1|1<2=2
a 1=1
a n=g$(\s x->(x:s)!n)(a<$>[n-1,n-2..1])<$>[1,2]

Answer 2

Python, 137 bytes

def a(n,s=[0],r=lambda l:max([0]+filter(lambda i:l[-i:]==l[-i*2:-i],range(len(l))))):
 for _ in[0]*n:s+=[r(s+[0])>r(s+[1])]
 return-~s[n]

This solution is using 0-based indexing.

Answer 3

Jelly, 25 24 22 20 bytes

2 bytes thanks to Dennis.

2;€µḣJf;`€$ṪLµÞḢ
Ç¡Ḣ

Try it online!

A port of my answer in Pyth.

Ç¡Ḣ   Main chain

 ¡    Repeat for (input) times:
Ç         the helper chain
  Ḣ   Then take the first element



2;€µḣJf;`€$ṪLµÞḢ  Helper chain, argument: z

2;€               append z to 1 and 2, creating two possibilities
   µ         µÞ   sort the possibilities by the following:
    ḣJ                generate all prefixes from shortest to longest
       ;`€            append the prefixes to themselves
      f   $           intersect with the original set of prefixes
           Ṫ          take the last prefix in the intersection
            L         find its length
                 Ḣ   take the first (minimum)

Answer 4

Mathematica, 84 bytes

a@n_:=a@n=First@MinimalBy[{1,2},Array[a,n-1]~Append~#/.{___,b___,b___}:>Length@{b}&]

Answer 5

Jelly, 30 29 28 27 24 bytes

;€¬;\€Z;/f;`€$ṪḢ;
1Ç¡o2Ḣ

Try it online! or verify all test cases.

Answer 6

MATL, 34 bytes

vXKi:"2:"K@h'(.+)\1$'XXgn]>QhXK]0)

Try it online! or verify all test cases.

Explanation

v             % Concatenate stack vertically: produces empty array
XK            % Copy to clipboard K. This clipboard holds the current sequence
i:            % Take input n. Generate vector [1 2 ... n]
"             % For each k in [1 2 ... n]
  2:          %   Push [1 2]. These are the possible digits for extending the sequence
  "           %     For each j in [1 2]
    K         %       Push contents of clipboard K (current sequence)
    @         %       Push j (1 or 2)
    h         %       Concatenate horizontally: gives a possible extension of sequence
    '(.+)\1$' %       String to be used as regex pattern: maximal-length repeated suffix
    XX        %       Regex match
    gn        %       Convert to vector and push its length: gives length of match
  ]           %    End. We now have the suffix lengths of the two possible extensions
  >           %    Push 1 if extension with "1" has longer suffix than with "2"; else 0 
  Q           %    Add 1: gives 2 if extension with "1" produced a longer suffix, or 1
              %    otherwise. This is the digit to be appended to the sequence
  h           %    Concatenate horizontally
  XK          %    Update clipboard with extended sequence, for the next iteration
]             % End
0)            % Get last entry (1-based modular indexing). Implicitly display

Answer 7

Python 2, 94 bytes

import re
s='1'
exec"s+=`3-int(re.search(r'(.*)(.)\\1$',s).groups()[1])`;"*input()
print s[-1]

Uses 0-based indexing. Test it on Ideone.

Answer 8

Pyth, 26 bytes

huh.mleq#.<T/lT2._b+RGS2QY

Test suite.

Explanation

When n = 6, we choose between 1 2 1 1 2 1 and 1 2 1 1 2 2.

We generate these two possibilities, and then look at their suffixes.

For the first one, the suffixes are: 1, 2 1, 1 2 1, 1 1 2 1, 2 1 1 2 1, 1 2 1 1 2 1.

We filter for doubled suffixes by checking if they are the same after rotating them for their length divided by 2 (it turns out that this check is not perfect, because it confirms 1 and 2 also).

We take the last doubled suffix and then take its length.

We then choose the possibility that corresponds to a minimum length generated above.

Then we proceed to the next value of n.

For the purpose of this program, it was golfier to generate the reversed array instead.

huh.mleq#.<T/lT2._b+RGS2QY
 u                      QY   repeat Q (input) times,
                             start with Y (empty array),
                             storing the temporary result in G:
                   +RGS2         prepend 1 and 2 to G,
                                 creating two possibilities
   .m             b              find the one that
                                 makes the following minimal:
                ._                   generate all prefixes
       q#                            filter for prefixes as T
                                     that equals:
         .<T/lT2                         T left-rotated
                                         by its length halved
      e                              take the last one
     l                               generate its length
  h                              take the first minimal one
h                                take the first one from the generated
                                 array and implicitly print it out

Answer 9

Pyth, 46 29 bytes

Took some inspiration from @Leaky Nun's excellent Pyth answer. Tried to see if there was a shorter way using strings. Still 3 bytes short!

huh.melM+kf!x>blTT._bm+dGS2Qk

You can try it out here.

Answer 10

Retina, 51 42 bytes

9 bytes thanks to Martin Ender.

.+
$*x
+`((.*)(^|2|(?<3>1))\2)x
$1$#3
!`.$

Try it online!

A port of this answer.

Answer 11

Perl, 40 bytes

$a.=/(.*)(.)\1$/^$2for($a)x$_;$_=$a%5+1

The code is 39 bytes long and requires the -p switch (+1 byte).

The loop is inspired by the Perl solution on the relevant OEIS page, although I did come up independently with the regular expression.

Test it on Ideone.

Answer 12

JavaScript (ES6), 84

Index base 0

n=>eval("s='1';for(r=d=>(s+d).match(/(.*)\\1$/)[0].length;n--;s+=c)c=r(1)>r(2)?2:1")

Less golfed

n=>{
  r = d => (s+d).match(/(.*)\1$/)[0].length;
  c = '1';
  for(s = c; n--; s += c)
    c = r(1) > r(2) ? 2 : 1;
  return c;
}

Test

F=
n=>eval("s='1';for(r=d=>(s+d).match(/(.*)\\1$/)[0].length;n--;s+=c)c=r(1)>r(2)?2:1")

for(n=0;n<20;n++)console.log(n,F(n))

Answer 13

Husk, 20 bytes

!¡λ◄(L←SnmDṫ:⁰)ḣ2);1

Try it online! or Verify first 20 values

Same algorithm as Leaky Nun's answers.