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Yes and no. The only square number that this does not work for is 1(since 0 is neither a square or a triangular number). It does however work for every other square number.

Unfortunately I do not know how to prove it mathematically.

Triangle Numbers 1 - 110

1 ,3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105

Square Numbers 1-200

1, 4(1+3), 9(3+6), 16(6+10), 25(10+15), 36(15+21), 49(21+28), 64(28+36), 81(36+45), 100(45+55), 121(55+66), 144(66+78), 169(78+91), 196(91+105)

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Q: Is every square number a sum of two consecutive triangular numbers?
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