Your calculation is wrong. The total is 9.2234 E18 (or 2E63). It would be right if there had been 2 grains in the first square, but there was only one.

Think again. Let's reduce 64 to 4, the number of squares filled in the illustration. 2^4 = 16 there are 1+2+4+8 = 15 grains on the 4 squares i.e. (2^4)-1 grains. This can be extended to n=64 and yield (2^64)-1 grains

@alain so funny. had there been 2 grains in first square, then there would be 2^64 in the 64th square. then how would the total be 2^64 - 1 !

What Alain has demonstrated is the power that an off-by-one error can have, especially at the wrong end of a series.