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That's an interesting website, which would seem to be a reasonable basis for your outlook on this topic.
However, there seems to be a problem.
The formula it provides for the "rule" of operation is always the same formula, regardless of the numbers chosen for the series.
.
I tried to change the 6th number from 19 to 13, and the site offered 139 as the 7th, giving the same formula.
.
I tried changing the 5th number from 1 to 17 (1,12,1,1,17,...) and the site offered 26 as the 6th, giving the same formula.
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I tried changing the 6th number from 26 to 1 (1,12,1,1,17,1,...) and the site gave -173 as the 7th, giving the same formula.
.
No matter what numbers I provide, it guesses the wrong number for the next in the series.
But as soon as I correct the wrong number, it provides another wrong number for the subsequent number following.
And in all cases, the same exact formula is provided as the "rule" that determines the next number.
.
So it's not working.
.
Try putting in 1,12... and asking for the third number. The site says that
23 is the third number.
Wrong: it should be
1.
Try putting in 1,12,1,... and ask for the fourth. It says
-32 is the fourth.
WRONG: it should be
1.
Try putting in 1,12,1,1,... and ask for the fifth. It says
45 is the fifth.
WRONG: it should be
1.
Try putting in 1,12,1,1,1,... and ask for the sixth. It says
-54 is the sixth.
WRONG: it should be the one I asked for in the puzzle.
We already tried 1,12,1,1,1,19,... and for the next, received
175.
We already tried 1,12,1,1,1,13,... and for the next, got
139.
We already tried 1,12,1,1,1,17,... and for the next, got
26.
.
And the real kicker is,
FOR ALL THESE WRONG NEXT NUMBERS AND THEIR CORRECTIONS, THE FORMULA REMAINS THE SAME.
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So just for kicks, let me give you the correct 6th number, which is 2. The site has been unable to provide that number.
Now, given the correct series from the puzzle, 1,12,1,1,1,2,... what does the site give as the formula?
That's right. It provides the same formula as it did for 1,12,1,1,1,... followed by 19, 13 and 17, all of which were
WRONG.
.
What does the site provide now for the 7th number in the series after 1,12,1,1,1,2,...? It says
73, which is
WRONG.
This site, as interesting as it is, has been UNABLE to correctly anticipate ANY of the numbers in this series, even the THIRD (1).
And it gives
the same formula FOR EVERYTHING, which by all reasonable expectations,
must also be WRONG.
This formula has exactly NOTHING to do with the very simple fact which correctly predicts all the numbers in the puzzle series.
.
Formula used:
Where, x and y are the coordinates
.
The site offers calculation of Lagrange polynomials.
Apparently what this puzzle asks for has nothing to do with Lagrange polynomials.
So their formula doesn't work on this puzzle.
If Lagrange polynomials are for constructing
curves, perhaps it would help to know that the puzzle is not based on any curve.