Interpolation error between even and odd FFT

Upsampling a spike [100 0 0 ...] with rfft - irfft gives quite different results for input lengths even or odd. Here's a plot for 100 and 101:

Why the difference ?
An intuitive explanation which could predict this on paper, without running it, would be nice.

The curve for odd $\to $ 2*odd goes $\sim 0, 1/n, 0, -1/n, 0 ...$ in the middle; that for even $\to$ 2*even approaches 0 in the middle (quadratically ?)

See scipy.signal.resample . The code in resample .py appears to do:

m2 = n // 2 + 1  # number of relevant frequency bins of a one-sided FFT
X = rfft(x) [:m2]
if n is even:  # Account for unpaired bin at m//2:
    X[n//2] *= 0.5
0-pad  # inside irfft
irfft

#!/usr/bin/env python3
"""Upsample a spike [100 0 0 ...] with periods even / odd
rfft, 0-pad, irfft (scipy.signal.resample)
"""
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.resample.html
# https://github.com/scipy/scipy/blob/main/scipy/signal/_signaltools.py

import numpy as np
from scipy.signal import resample
import matplotlib.pyplot as pl
import sys

print( 80 * "▄" )
np.set_printoptions( edgeitems=10, threshold=10, linewidth=120, formatter = dict(
        float = lambda x: "0" if abs(x) < 1e-10  else "%.2g" % x ))  # 2 digits, tiny -> 0

def spike( n ):
    x = np.zeros( n )
    x[0] = 100
    return x

#...............................................................................
periods = [100, 101]
zoom = 2
plot = 0
    # to change these params, run this.py  a=1  b=2,None  'c = expr' ... in sh or ipython --
for arg in sys.argv[1:]:
    exec( arg )
print( f"{periods = }  {zoom = } " )

title = __doc__
print( title, "\n" )
if plot:
    fig, axes = pl.subplots( nrows=len(periods), figsize=[8, 5] )
    fig.suptitle( title, multialignment="left" )
else:
    axes = len(periods) * [None]

for period, ax in zip( periods, axes ):
    x = spike( period )
    nout = zoom * period
#...............................................................................
    y = resample( x, nout )
    jmid = nout // 2
    ymid = y[jmid-5 : jmid+5+1]
    print( "period %d  y middle * 100: %s " % ( period, ymid * 100 ))
    if plot:
        title = r"$ %d \to %d $" % ( period, nout )
        ax.set_title( title, loc="center", y=0.9, fontsize=12 )
        ax.set( ylim=[-2, 2] )
        ax.plot( y, "s-", lw=0.36, ms=1, color="crimson" )
"""
period 100  y middle * 100: [-7.9  0 4.7   0 -1.6   0  -1.6   0 4.7  0 -7.9]
period 101  y middle * 100:    [0 99   0 -99    0  99     0 -99   0 99    0]
"""