How to write a pattern that matches all linear combinations of functions involved except 0?

Question

I am trying to write the simplest (and efficient) pattern that matches the linear combination of all functions except when the combination produces zero?

Consider the following as an example:

pattern=HoldPattern[Except[0,(_:1)*Optional[Fibonacci[_],0]+(_:1)*Optional[LucasL[_],0]]];

Now if I use this to match the following cases:

MatchQ[pattern]/@{
    0,
    2 LucasL[n],
    Fibonacci[n],
    -(Fibonacci[n]/2)+LucasL[n]/2
}

I want the pattern to produce true for the last three, i.e. {False, True, True, True}.

How do I fix this pattern?

Answer 1

There is no Optional Head Pattern specification that I know of. In the OP pattern the Plus head is specified in the pattern so cases of individual terms are not matched.

I use Alternatives to construct a pattern for the terms, termPattern. Then use Alternatives with Repeated to allow for an individual term or a linear combination of the terms.

With

termPattern = (_ : 1) Optional[Fibonacci[_] | LucasL[_], 0];
pattern = Except[0, termPattern | HoldPattern[Plus][termPattern ..]];

Then

MatchQ[pattern]/@{
    0,
    2 LucasL[n],
    Fibonacci[n],
    -(Fibonacci[n]/2) + LucasL[n]/2
}

{False, True, True, True}

Also

MatchQ[pattern][
 a LucasL[n] + b Fibonacci[n] - (Fibonacci[n]/2) + LucasL[n]/2
 ]

True

Hope this helps.