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I noticed the the language L generates words with a length that represents trianglar numbers: 1,3,6,10,15 etc..

I'm trying to use the pempingt lemma for w=1^(p(p+1) but I didnt reach anywhere..

Can someone help or give me an idea how to solve it ?

Thanks ! :)

Sofiene Fehri
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All context-free languages over one-letter alphabet are regular. So if L was context-free it would also be regular. All regular languages over one-letter alphabet are ultimately periodic, which your L is not, because the gaps get all the time bigger.

Or with the pumping lemma: any pumping results in a language uv^i w x^i y. Because all letters are the same we can exchange factors, and this is equal to uyw v^i x^i = uyw(vx)^i. In L the distance between one word and the next will get bigger than any |vx|^i as n grows.

Peter Leupold
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