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Double Product in the proof of partitions of $n$
So I am supposed to show that the number of partitions of $n$ for which no part appears more than twice is equal to the number of partitions of n for which no part is divisible by 3. and there is one ...
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# of partitions of $n$ into at most $r$ positive integers $=$ # of partitions of $n + r$ into exactly $r$ positive integers?
How do I see that the number of partitions of the integer $n$ into at most $r$ positive integers is equal to the number of partitions of $n + r$ into exactly $r$ positive integers?
user223957
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Find a recurrence for the number of integer compositions of n which only have 1s and 2s as parts
Find a recurrence for $$i_n$$ the number of integer compositions of $n$ which only have $1$s and $2$s as parts.
How do you approach this problem?
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