'Why this shift operation plus bitwise works only up to 31?

Why shift operation below works and end up equal? Is there any name for this pattern? I am trying to find out what was going on in the head of the person who wrote this code!

int i = 0x1;
i |= 0x1 << 1;
i |= 0x1 << 2;
i |= 0x1 << 3;
i |= 0x1 << 4;
i |= 0x1 << 5;

int j = 5;

if( ((0x1 << (j + 1)) - 1) == i)
{
    // WHY?
}

I tried to check if this is true for all numbers but only works up to 31.

for (int i = 1; i <= 100; i++) {
    int total_1 = 0x1;
    for (int j = 1; j <= i; j++) {
        total_1 |= 0x1 << j;
    }

    int total_2 = (0x1 << (i + 1)) - 1;
    if (total_2 == total_1) {
    } else {
        cout << i << endl;
        break;
    }
}

UPDATE

Please explain the first part why they would end up equal?

c++cbit-shift


Solution 1:[1]

UPDATE Please explain the first part why they would end up equal?

((0x1 << (j + 1)) - 1) sets j lower bits to 1

Example:

If j is 3, j+1 is 4, 1 shifted by 4 is 0b1000; subtract 1 - you get 0b0111

Re: Is there any name for this pattern? - I looked here, this pattern is used a couple of times, but is not named. Probably, too obvious :)

Sources

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Source: Stack Overflow

Solution Source
Solution 1

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