Differationale
Dedicated to Alan Ray Niquette 1943-2014

olvers who enjoy number-challenges will like the On-Line Encyclopedia of Integer Sequences (OEIS), where one finds dozens of myriads (literally, more than a few x 12 x 10,000) sequences collected into a searchable database.  One may find something else, too -- an apparent assumption about the meaning of the word solution as it pertains to integer sequences.  Let us make that assumption explicit here, but first...
Nota bene, in the Next Number puzzle, solvers were given a simple question...

 1, 15, 46, 100, 183, 301, 460, ... What is the next integer?
...such that for the given integers the solution is merely one integer -- the "next number."  OEIS goes beyond that, requesting and providing algorithm(s) for the derivation of all the given integers plus the unknown next number.

t goes without say -- oh well, let's say it anyway -- every solution of such a puzzle requires a method, often an algorithm.  Every such algorithm must take into consideration all of the given integers. OEIS does not restrict the algorithm to use only the given integers.  Indeed, in postulating an algorithm, one is free to use other numerical sources.
For example, the index of each given number in the series (i) frequently appears in a solution.  Each number n(i) in the series can be represented, say, as a function of i or as a function of some previous number n(i-k).  Indeed, all kinds of numbers can be brought to the party and invoked by an algorithm, like prime numbers, binomial expansions, factorials, Fibonacci numbers, whatever.  At the extreme, one finds  that this series: 2, 15, 1001, 215441 ... is the "product of the next i prime numbers."  Ugh.
Nota bene, the Next Number puzzle did not indulge in such liberties.  The solution used only the given integers with no regard for how those integers were formed.  The given integers were manipulated only arithmetically, using a deterministic algorithm which is herein named Differationale
Excuse my French, but the word differationale is an anglicized portmanteau, combining différence, an arithmetic term,  and raisonnement a reasoning term.
As an algorithm, differationale will be found to have immense power.  It provides a solution to any next-number puzzle throughout the known universe and in perpetuity.  The expression "throughout the known universe and in perpetuity" may appear a bit extravagant. However, as we shall see, the claim is fully justified by the word puzzle.

 Every puzzle is the creation of a "puzzle-master" (from the French  maître-puzzle, a gender-neutral expression).  It is incumbent upon the puzzle-master to formulate both the puzzle and its solution -- plus a rationale for the solution.  Obviously, the puzzle-solveur must respond with a solution -- plus a rationale for that solution.

he puzzle-solver's solution may or may not agree with the puzzle-master's solution. Which one is "right" and which one is "wrong"?  To illustrate that question, here is a collection of next-number puzzles created by a certain maître-puzzle...

 Puzzle: 0 1 2 3 4 5 6 7 8 9 10 First 1 1 1 1 1 1 1 1 1 1 1 15 2 3 4 8 32 64 2 3 4 101 46 4 9 9 27 243 729 6 6 10 202 100 8 27 16 64 1,024 4,096 24 10 20 303 183 16 81 25 125 3,125 15,625 120 15 35 404 301 32 243 36 216 7,776 46,656 720 21 56 505 Last 460 64 729 49 343 16,807 117,649 5,040 28 84 606 Next ? ? ? ? ? ? ? ? ? ? ?