1 6 26 81 what next number?

1 6 26 81 ?

The factors of 26 are 1*26 or 2*13, so the series isn’t multiplicative…

The differences are 5, 20, and 55, so additive is out….

It may just be a coincidence, but 6 = 3×1 + 3 and 81 = 3×26 + 3.
Unfortunately 3×6 + 3 = 21, not 26.

Not it

double post ^^
Should be 156

626
Here is my reasoning:
The sequence is: 16268
The first two numbers contained one digit, and the next two contained two digits. The next two numbers will contain three digits and follow the sequence of numbers: 626, 816

156…the sum of the differences of the differences of the differences.

1 6 26 81 (156)
5 20 55 (75)
15 35 (55)
20 (20)

so 1 plus the dif (5) =6
6 plus the dif (20) =26
26 plus the dif (55) =81 now go down to the next level…
5 plus the dif (20) =20
20 plus the dif (35) = 55 now go down to the next level and we see the difference is 20 and use that to extend out the sequence and if we use 20 as the difference between the next =level above….30 + 20 =55 and then add 20 + 55 to gain the number we need of 75 to complete the sequence and find our next number of 81+75 = 156

I tried looking in an online sequence library and was told that there was no match. My guess is that the sequence is based on something that is not numeric.

@Cruiser , What you did, whether you realize it or not, was to find the smallest polynomial that starts out with those values. You could do that with any set of values, so I would guess that that is not the answer.

A search of the Online Encyclopedia of Integer Sequences turned up no results. (The database contains over 200,000 sequences). Edit: It seems that @LostInParadise beat me to it.

Anyways, I don’t feel that there is a meaningful connection between the numbers. Given only four numbers, there may be many patterns which might not pan out.

@LostInParadise Whether it is the answer or not (@ARE_you_kidding_me offered the same number) it seems to work for that sequence and works quite well if you continue that pattern of adding the differences and it was the most fun I have had with math in a long time! ;) Using my fuzzy math the next number would be 251

I know I am going to hate myself for pointing this out, but the 75 in your table should be 110, since 55+55 = 110. That would make the next number in the sequence equal to 110+81= 191.

The corresponding polynomial is:
1 + 5n + (15/2)n(n-1) + (20/6)n(n-1)(n-2), where 1, 5 15 and 20 come from the first column of the table.

If you plug in n=0, 1, 2 3 and 4 you will get 1, 6, 26, 81 and 191.

Something Newton thought of on the way to discovering calculus.

I agree that 191 is correct if the cubic regression is the approach you want to take:
f(X) = 10/3 * X^3 – 2.5 * X^2 + 25/6 * X + 1

Here are my attempts at a recurrence relation:

f(X) = 15/2 * f(X-1) – 19 * f(X-2)
non-integer result

f(X) = 9/2 * f(X-1) – f(X-2)^2
non-integer result

f(X) = f(X-1) + 7 * f(X-2) + 13
gives 276

f(X) = f(X-1) + f(X-2) ^ 2 + 19
gives 776

In my opinion, the last two are much more “interesting” than the cubic regression, as they contain only integers (but also only a few), yet still give a meaningful result.