Here is pifricted's neiborhood ideas.
From T=3n+0 to T=3n+2:
x=11, y = 4, rule = B/S0123456HHistory
5.F3.F$.A3.AF2.FA$.2F!
# [[ COLOR 1 255 255 255 VIEWONLY GRID ]]From T=6n+0 to T=6n+5:
x=24, y = 4, rule = B/S0123456HHistory
8.F3.2F3.F$.A2.FA2.FA3.A3.AF3.AF$.2F2.F17.F!
# [[ COLOR 1 255 255 255 VIEWONLY GRID ]]From T=2n+0 to T=2n+1:
x=30, y = 9, rule = B/S0123456HHistory
D.A.D.A9.D.A.D.A$.2D2A2D2A6.2D2A2D2A$.DFACDFAC8.DFACDFAC$3.2F2C2F2C6.
2F2C2F2C$3.FDCAFDCA8.FDCAFDCA$5.2D2A2D2A6.2D2A2D2A$5.DFACDFAC8.DFACDF
AC$7.2F2C2F2C6.2F2C2F2C$7.F.C.F.C9.F.C.F.C!
!
# [[
COLOR 1 127 127 127
COLOR 4 167 167 167
VIEWONLY GRID ]]Alternation
x=3, y = 3, rule = B3/S3HHistory
AB$BCA$.AB!
# [[ COLOR 1 127 127 127
AUTOSTART
GPS 3 GRID ]]From T=3n+0 to T=3n+2:
x=31, y = 8, rule = B/S0123456HHistory
.8B3.8B3.8B$.2BCBA3B3.8B3.8B$.B2C2A3B3.2BCBA3B3.3BCBA2B$.2BECDA2B3.B2C
2A3B3.2B2C2A2B$.B2E2D3B3.2BECDA2B3.3BECDAB$.3BEBD2B3.B2E2D3B3.2B2E2D2B
$.8B3.3BEBD2B3.4BEBDB$.8B3.8B3.8B!
# [[
COLOR 1 127 127 127
COLOR 4 167 167 167
COLOR 5 137 167 137
VIEWONLY GRID ]]