Ramanujan results for middle school?

You can try the Rogers–Ramanujan identities:

• The number of partitions of $n$ in which adjacent parts are at least 2 apart is the same as the number of partitions of $n$ in which each part ends with 1,4,6,9.
• The number of partitions of $n$ without 1 in which adjacent parts are at least 2 apart is the same as the number of partitions of $n$ in which each part ends with 2,3,7,8.

For example, taking $n=10$:

• Partitions in which adjacent parts are at least 2 apart: $$ 10 = 10 = 9 + 1 = 8 + 2 = 7 + 3 = 6 + 4 = 6 + 3 + 1 $$
• Partitions in which each part ends with 1,4,6,9: $$ 10 = 9 + 1 = 6 + 4 = 6 + 1 + 1 + 1 + 1 = 4 + 4 + 1 + 1 = 4 + 6times 1 = 10 times 1 $$
• Partitions without 1 in which adjacent parts are at least 2 apart: $$ 10 = 10 = 8 + 2 = 7 + 3 = 6 + 4 $$
• Partitions in which each part ends with 2,3,5,8: $$ 10 = 8 + 2 = 7 + 3 = 3 + 3 + 2 + 2 = 2 + 2 + 2 + 2 + 2 $$