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I got

sage: n=var('n')
sage: assume(n>=0)
sage: assume(n,'integer')
sage: a_n=1/20*(1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) + 25) - 1/20*(-1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) - 25) - 1/12*
....: (1/2*I*sqrt(3) + 1/2)^n*(I*sqrt(3) + 3) - 1/12*(-1/2*I*sqrt(3) + 1/2)^n*(-I*sqrt(3) + 3)
sage: a_n.real_part()
------------------------------------------------------------------------
Unhandled SIGSEGV: A segmentation fault occurred.
This probably occurred because a *compiled* module has a bug
in it and is not properly wrapped with sig_on(), sig_off().
Python will now terminate.
------------------------------------------------------------------------
Segmentation fault (core dumped)

Will update my answer as I find out more, but obviously a segfault shouldn't happen here.

I got

sage: n=var('n')
sage: assume(n>=0)
sage: assume(n,'integer')
sage: a_n=1/20*(1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) + 25) - 1/20*(-1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) - 25) - 1/12*
....: (1/2*I*sqrt(3) + 1/2)^n*(I*sqrt(3) + 3) - 1/12*(-1/2*I*sqrt(3) + 1/2)^n*(-I*sqrt(3) + 3)
sage: a_n.real_part()
------------------------------------------------------------------------
Unhandled SIGSEGV: A segmentation fault occurred.
This probably occurred because a *compiled* module has a bug
in it and is not properly wrapped with sig_on(), sig_off().
Python will now terminate.
------------------------------------------------------------------------
Segmentation fault (core dumped)

Will update my answer as I find out more, but obviously a segfault shouldn't happen here.

I got

sage: n=var('n')
sage: assume(n>=0)
sage: assume(n,'integer')
sage: a_n=1/20*(1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) + 25) - 1/20*(-1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) - 25) - 1/12*
....: (1/2*I*sqrt(3) + 1/2)^n*(I*sqrt(3) + 3) - 1/12*(-1/2*I*sqrt(3) + 1/2)^n*(-I*sqrt(3) + 3)
sage: a_n.real_part()
------------------------------------------------------------------------
Unhandled SIGSEGV: A segmentation fault occurred.
This probably occurred because a *compiled* module has a bug
in it and is not properly wrapped with sig_on(), sig_off().
Python will now terminate.
------------------------------------------------------------------------
Segmentation fault (core dumped)

Will update my answer as I find out more, but obviously a segfault shouldn't happen here.

Confirmed on Linux and Windows. The segfault is coming from pynac.pynac. It seems to be specifically the assume(n, 'integer') part that does it. Without that it doesn't crash.

I got

sage: n=var('n')
sage: assume(n>=0)
sage: assume(n,'integer')
sage: a_n=1/20*(1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) + 25) - 1/20*(-1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) - 25) - 1/12*
....: (1/2*I*sqrt(3) + 1/2)^n*(I*sqrt(3) + 3) - 1/12*(-1/2*I*sqrt(3) + 1/2)^n*(-I*sqrt(3) + 3)
sage: a_n.real_part()
------------------------------------------------------------------------
Unhandled SIGSEGV: A segmentation fault occurred.
This probably occurred because a *compiled* module has a bug
in it and is not properly wrapped with sig_on(), sig_off().
Python will now terminate.
------------------------------------------------------------------------
Segmentation fault (core dumped)

Will update my answer as I find out more, but obviously a segfault shouldn't happen here.

Confirmed on Linux and Windows. The segfault is coming from pynac. It seems to be specifically the assume(n, 'integer') part that does it. Without that it doesn't crash.

sage: n=var('n')
sage: assume(n, 'integer')
sage: (I^n).real_part()
<segfault>

src/sagemath/sage/local/lib/libpynac.so.18(_ZNK5GiNaC5power9real_partEv+0x51)[0x7f7c45607bf1]

sage: n=var('n')
sage: (I^n).real_part()
cos(1/2*pi*real_part(n))*e^(-1/2*pi*imag_part(n))

I got

sage: n=var('n')
sage: assume(n>=0)
sage: assume(n,'integer')
sage: a_n=1/20*(1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) + 25) - 1/20*(-1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) - 25) - 1/12*
....: (1/2*I*sqrt(3) + 1/2)^n*(I*sqrt(3) + 3) - 1/12*(-1/2*I*sqrt(3) + 1/2)^n*(-I*sqrt(3) + 3)
sage: a_n.real_part()
------------------------------------------------------------------------
Unhandled SIGSEGV: A segmentation fault occurred.
This probably occurred because a *compiled* module has a bug
in it and is not properly wrapped with sig_on(), sig_off().
Python will now terminate.
------------------------------------------------------------------------
Segmentation fault (core dumped)

Will update my answer as I find out more, but obviously a segfault shouldn't happen here.

Confirmed on Linux and Windows. The segfault is coming from pynac. It seems to be specifically the assume(n, 'integer') part that does it. Without that it doesn't crash.

I spent a few minutes trying to narrow this down to a simpler example and got this

sage: n=var('n')
sage: assume(n, 'integer')
sage: (I^n).real_part()
<segfault>

It appears to go into an infinite recursion and eventually overflow the stack in

src/sagemath/sage/local/lib/libpynac.so.18(_ZNK5GiNaC5power9real_partEv+0x51)[0x7f7c45607bf1]

The correct answer for $\mathfrak{Re}[i^n]$ should be $\cos\left(\frac{\pi n}{2}\right)$ but I guess it's not that clever, and it certainly shouldn't crash...

If I don't stipulate that $n \in \mathbb{Z}$ then it does give the correct answer:answer, sort of...

sage: n=var('n')
sage: (I^n).real_part()
cos(1/2*pi*real_part(n))*e^(-1/2*pi*imag_part(n))

I got

sage: n=var('n')
sage: assume(n>=0)
sage: assume(n,'integer')
sage: a_n=1/20*(1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) + 25) - 1/20*(-1/2*sqrt(5) + 1/2)^n*(11*sqrt(5) - 25) - 1/12*
....: (1/2*I*sqrt(3) + 1/2)^n*(I*sqrt(3) + 3) - 1/12*(-1/2*I*sqrt(3) + 1/2)^n*(-I*sqrt(3) + 3)
sage: a_n.real_part()
------------------------------------------------------------------------
Unhandled SIGSEGV: A segmentation fault occurred.
This probably occurred because a *compiled* module has a bug
in it and is not properly wrapped with sig_on(), sig_off().
Python will now terminate.
------------------------------------------------------------------------
Segmentation fault (core dumped)

Will update my answer as I find out more, but obviously a segfault shouldn't happen here.

Confirmed on Linux and Windows. The segfault is coming from pynac. It seems to be specifically the assume(n, 'integer') part that does it. Without that it doesn't crash.

I spent a few minutes trying to narrow this down to a simpler example and got this

sage: n=var('n')
sage: assume(n, 'integer')
sage: (I^n).real_part()
<segfault>

It appears to go into an infinite recursion and eventually overflow the stack in

src/sagemath/sage/local/lib/libpynac.so.18(_ZNK5GiNaC5power9real_partEv+0x51)[0x7f7c45607bf1]

The correct answer for $\mathfrak{Re}[i^n]$ should be $\cos\left(\frac{\pi n}{2}\right)$ but I guess it's not that clever, and it certainly shouldn't crash...

If I don't stipulate that $n \in \mathbb{Z}$ then it does give the correct answer, sort of...

sage: n=var('n')
sage: (I^n).real_part()
cos(1/2*pi*real_part(n))*e^(-1/2*pi*imag_part(n))