difference equations - Are there different Method options that can be used with FindSequenceFunction?

The help page states that Method is an option to use with FindSequenceFunction, but I don't find any further information. (There is also a FunctionSpace option--for which I know that "Polynomial" is one possibility. But the full list is not easy, it seems, to locate. I presume that if a FunctionSpace option is not specified, all possibilities are explored.)

Answer

This is not a complete answer. It is a long comment about what I found when I looked at the inspectable source code of this function using various spelunking tools (e.g.. Spelunking package, GeneralUtilities`PrinteDefinitions, etc.).

Method parsing looks unusual

First, I wonder if parsing for the Method option is buggy. First it does this internally:

$UserMethod = Method /. Flatten[{opts}] /. Options[FindSequenceFunction];

Then the whole function has a condition (/;) on evaluation which includes this:

($UserMethod===Automatic||OptionQ[$UserMethod])

To translate what this means: the only accepted Method specifications are Method -> Automatic or something of the form Method -> (opt -> value). This double Rule construct could be what was meant, but is definitely suspect.

Further evidence:

FindSequenceFunction[{1, 1, 2, 3, 5, 8, 13}, n, Method -> "foo"]
(* FindSequenceFunction[{1, 1, 2, 3, 5, 8, 13}, n, Method -> "foo"] *)

FindSequenceFunction[{1, 1, 2, 3, 5, 8, 13}, n, Method -> (foo -> "foo")]
(* Fibonacci[n] *)

I wasn't able to dig deep enough to discover concrete accepted values for Method

Values for FunctionSpace

Automatic and All are equivalent, I think.

Other than these, try any of the following or a list of them:

"PeriodicSequence"
"Polynomial"
"RationalFunction"
"HypergeometricTerm"
"ConstantRecursive"
"HolonomicSequence"
"QHolonomicSequence"

Values for TransformationFunctions

A list of some of these:

"Ratio", "Difference", "BinomialTransform", 
"BinomialInverseTransform", "BoustrophedonTransform", 
"BoustrophedonInverseTransform", "StirlingTransform", 
"StirlingInverseTransform", "MoebiusTransform", 
"MoebiusInverseTransform", "Convolution", "ExponentialConvolution"

You can also use All, which means all of them, or Automatic, which means {"Ratio", "Difference"}.

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Warning: All this information is obtained by looking naively at the implementation. These are just option values which I think are valid. I am not certain, I may have made a mistake. I don't know what the effect of these values are. Since they are undocumented, I don't know if they are meant to be used at all. Some may be experimental or unimplemented. Don't be surprised if Mathematica does unexpected things when you use them.