Find the pattern

I feel tired to do "find the pattern" exercise such as

1 2 3 4 (5)
1 2 4 8 (16)
1 2 3 5 8 (13)


Please write a program that finds the pattern for me.

Here, we define the pattern as a recurrence relation that fits the given input, with the smallest score. If there are multiple answers with the same smallest score, using any one is fine.

Let the $$\k\$$ first terms be initial terms (for the recurrence relation, etc.), and the $$\i\$$'th term be $$\f(i)\$$ ($$\i>k,i\in\mathbb N\$$).

• A non-negative integer $$\x\$$ adds$$\\lfloor\log_2\max(|x|,1)\rfloor+1\$$ to the score
• The current index $$\i\$$ adds $$\1\$$ to the score
• +, -, *, / (round down or towards zero, as you decide) and mod (a mod b always equal to a-b*(a/b)) each add $$\1\$$ to the score
• For each initial term $$\x\$$, add $$\\lfloor\log_2\max(|x|,1)\rfloor+2\$$ to the score
• $$\f(i-n)\$$ (with $$\n\le k\$$) adds $$\n\$$ to the score. E.g. Using the latest value $$\f(i-1)\$$ add $$\1\$$ to the score, and there must be at least 1 initial term.
• Changing the calculation order doesn't add anything to the score. It's fine if you write 1+i as 1 i +, +(i,1) or any format you like.

Samples:

input   -> [score]    expression
1 2 3 4     -> [1]    f(i) = i
1 2 4 8     -> [5]    f(1) = 1, f(i) = f(i-1)+f(i-1)
1 2 3 5 8   -> [9]    f(1) = 1, f(2) = 2, f(i) = f(i-1)+f(i-2)


Shortest program in each language wins. It's fine if your program only solve the problem theoretically.

• Should 1, 2, 4, 8 be f(1) = 1, f(i) = f(i - 1) + f(i - 1)? And how current score 5 got? – tsh Oct 29 '18 at 10:13
• f(1) = 1 is 2 points, f(2) = 2 is 3 points, f(i-1) is 1 point, + is 1 point, f(i-2) is 2 points. So f(1) = 1, f(2) = 2, f(i) = f(i-1) + f(i-2) is 9 points. Where am I wrong? – tsh Oct 29 '18 at 10:42
• @tsh Fixed issues. Data generated by hand – l4m2 Oct 29 '18 at 11:17
• +1 how is the 1,2,4,8 scored at 5? I see an "initial term x" of 1 adding two, "the current index i" twice adding one each, "two non-negative integer x" both with value 1 each adding one, and a +adding one. This makes seven, but I could easily be misinterpreting something. – Jonathan Allan Oct 29 '18 at 11:27
• Does this answer your question? What comes next? – user85052 Jan 27 at 12:20