Gaps in the sequence, part 2

Posted on June 12, 2021 by Rich Holmes

I generated Euclidean and gap sequences for:

Sequence lengths $s$ from 8 through 16 steps
Event numbers $n$ from 3 through through length-3 (anything outside that range not being so interesting)
For gap, all generators $g \le s/2$ that can generate that many events (skipping those for which $s/\mathrm{gcd}(s,g) < n$ )

Here they are. Gap sequences that are the same as the corresponding Euclidean sequence (with possibly some offset) are tagged as such.

Steps 8 events 3
Euclid
       --O--O-O
Gap
gen  1 OOO----- 
gen  2 O-O-O--- 
gen  3 O--O--O- = Euclid

Steps 8 events 4
Euclid
       -O-O-O-O
Gap
gen  1 OOOO---- 
gen  2 O-O-O-O- = Euclid
gen  3 OO-O--O- 

Steps 8 events 5
Euclid
       -O-OO-OO
Gap
gen  1 OOOOO--- 
gen  3 OO-OO-O- = Euclid

Steps 9 events 3
Euclid
       --O--O--O
Gap
gen  1 OOO------ 
gen  2 O-O-O---- 
gen  3 O--O--O-- = Euclid
gen  4 O---O---O 

Steps 9 events 4
Euclid
       --O-O-O-O
Gap
gen  1 OOOO----- 
gen  2 O-O-O-O-- = Euclid
gen  4 O--OO---O 

Steps 9 events 5
Euclid
       -O-O-O-OO
Gap
gen  1 OOOOO---- 
gen  2 O-O-O-O-O = Euclid
gen  4 O--OO--OO 

Steps 9 events 6
Euclid
       -OO-OO-OO
Gap
gen  1 OOOOOO--- 
gen  2 OOO-O-O-O 
gen  4 O-OOO--OO 

Steps 10 events 3
Euclid
       ---O--O--O
Gap
gen  1 OOO------- 
gen  2 O-O-O----- 
gen  3 O--O--O--- = Euclid
gen  4 O---O---O- 

Steps 10 events 4
Euclid
       --O-O--O-O
Gap
gen  1 OOOO------ 
gen  2 O-O-O-O--- 
gen  3 O--O--O--O 
gen  4 O-O-O---O- 

Steps 10 events 5
Euclid
       -O-O-O-O-O
Gap
gen  1 OOOOO----- 
gen  2 O-O-O-O-O- = Euclid
gen  3 O-OO--O--O 
gen  4 O-O-O-O-O- = Euclid

Steps 10 events 6
Euclid
       -O-OO-O-OO
Gap
gen  1 OOOOOO---- 
gen  3 O-OO-OO--O 

Steps 10 events 7
Euclid
       -OO-OO-OOO
Gap
gen  1 OOOOOOO--- 
gen  3 O-OO-OO-OO = Euclid

Steps 11 events 3
Euclid
       ---O---O--O
Gap
gen  1 OOO-------- 
gen  2 O-O-O------ 
gen  3 O--O--O---- 
gen  4 O---O---O-- = Euclid
gen  5 O----O----O 

Steps 11 events 4
Euclid
       --O--O--O-O
Gap
gen  1 OOOO------- 
gen  2 O-O-O-O---- 
gen  3 O--O--O--O- = Euclid
gen  4 OO--O---O-- 
gen  5 O---OO----O 

Steps 11 events 5
Euclid
       --O-O-O-O-O
Gap
gen  1 OOOOO------ 
gen  2 O-O-O-O-O-- = Euclid
gen  3 OO-O--O--O- 
gen  4 OO--OO--O-- 
gen  5 O---OO---OO 

Steps 11 events 6
Euclid
       -O-O-O-O-OO
Gap
gen  1 OOOOOO----- 
gen  2 O-O-O-O-O-O = Euclid
gen  3 OO-OO-O--O- 
gen  4 OO--OO--OO- 
gen  5 O--OOO---OO 

Steps 11 events 7
Euclid
       -O-OO-OO-OO
Gap
gen  1 OOOOOOO---- 
gen  2 OOO-O-O-O-O 
gen  3 OO-OO-OO-O- = Euclid
gen  4 OOO-OO--OO- 
gen  5 O--OOO--OOO 

Steps 11 events 8
Euclid
       -OO-OOO-OOO
Gap
gen  1 OOOOOOOO--- 
gen  2 OOOOO-O-O-O 
gen  3 OO-OO-OO-OO 
gen  4 OOO-OOO-OO- = Euclid
gen  5 O-OOOO--OOO 

Steps 12 events 3
Euclid
       ---O---O---O
Gap
gen  1 OOO--------- 
gen  2 O-O-O------- 
gen  3 O--O--O----- 
gen  4 O---O---O--- = Euclid
gen  5 O----O----O- 

Steps 12 events 4
Euclid
       --O--O--O--O
Gap
gen  1 OOOO-------- 
gen  2 O-O-O-O----- 
gen  3 O--O--O--O-- = Euclid
gen  5 O--O-O----O- 

Steps 12 events 5
Euclid
       --O-O--O-O-O
Gap
gen  1 OOOOO------- 
gen  2 O-O-O-O-O--- 
gen  5 O--O-O--O-O- = Euclid

Steps 12 events 6
Euclid
       -O-O-O-O-O-O
Gap
gen  1 OOOOOO------ 
gen  2 O-O-O-O-O-O- = Euclid
gen  5 OO-O-O--O-O- 

Steps 12 events 7
Euclid
       -O-O-OO-O-OO
Gap
gen  1 OOOOOOO----- 
gen  5 OO-O-OO-O-O- = Euclid

Steps 12 events 8
Euclid
       -OO-OO-OO-OO
Gap
gen  1 OOOOOOOO---- 
gen  5 OO-O-OO-O-OO 

Steps 12 events 9
Euclid
       -OOO-OOO-OOO
Gap
gen  1 OOOOOOOOO--- 
gen  5 OO-OOOO-O-OO 

Steps 13 events 3
Euclid
       ----O---O---O
Gap
gen  1 OOO---------- 
gen  2 O-O-O-------- 
gen  3 O--O--O------ 
gen  4 O---O---O---- = Euclid
gen  5 O----O----O-- 
gen  6 O-----O-----O 

Steps 13 events 4
Euclid
       ---O--O--O--O
Gap
gen  1 OOOO--------- 
gen  2 O-O-O-O------ 
gen  3 O--O--O--O--- = Euclid
gen  4 O---O---O---O 
gen  5 O-O--O----O-- 
gen  6 O----OO-----O 

Steps 13 events 5
Euclid
       --O--O-O--O-O
Gap
gen  1 OOOOO-------- 
gen  2 O-O-O-O-O---- 
gen  3 O--O--O--O--O 
gen  4 O--OO---O---O 
gen  5 O-O--O-O--O-- = Euclid
gen  6 O----OO----OO 

Steps 13 events 6
Euclid
       --O-O-O-O-O-O
Gap
gen  1 OOOOOO------- 
gen  2 O-O-O-O-O-O-- = Euclid
gen  3 O-OO--O--O--O 
gen  4 O--OO--OO---O 
gen  5 O-O--O-O--O-O 
gen  6 O---OOO----OO 

Steps 13 events 7
Euclid
       -O-O-O-O-O-OO
Gap
gen  1 OOOOOOO------ 
gen  2 O-O-O-O-O-O-O = Euclid
gen  3 O-OO-OO--O--O 
gen  4 O--OO--OO--OO 
gen  5 O-O-OO-O--O-O 
gen  6 O---OOO---OOO 

Steps 13 events 8
Euclid
       -O-OO-O-OO-OO
Gap
gen  1 OOOOOOOO----- 
gen  2 OOO-O-O-O-O-O 
gen  3 O-OO-OO-OO--O 
gen  4 O-OOO--OO--OO 
gen  5 O-O-OO-O-OO-O = Euclid
gen  6 O--OOOO---OOO 

Steps 13 events 9
Euclid
       -OO-OO-OO-OOO
Gap
gen  1 OOOOOOOOO---- 
gen  2 OOOOO-O-O-O-O 
gen  3 O-OO-OO-OO-OO = Euclid
gen  4 O-OOO-OOO--OO 
gen  5 OOO-OO-O-OO-O 
gen  6 O--OOOO--OOOO 

Steps 13 events 10
Euclid
       -OOO-OOO-OOOO
Gap
gen  1 OOOOOOOOOO--- 
gen  2 OOOOOOO-O-O-O 
gen  3 OOOO-OO-OO-OO 
gen  4 O-OOO-OOO-OOO = Euclid
gen  5 OOO-OOOO-OO-O 
gen  6 O-OOOOO--OOOO 

Steps 14 events 3
Euclid
       ----O----O---O
Gap
gen  1 OOO----------- 
gen  2 O-O-O--------- 
gen  3 O--O--O------- 
gen  4 O---O---O----- 
gen  5 O----O----O--- = Euclid
gen  6 O-----O-----O- 

Steps 14 events 4
Euclid
       ---O--O---O--O
Gap
gen  1 OOOO---------- 
gen  2 O-O-O-O------- 
gen  3 O--O--O--O---- 
gen  4 O---O---O---O- 
gen  5 OO---O----O--- 
gen  6 O---O-O-----O- 

Steps 14 events 5
Euclid
       --O--O--O--O-O
Gap
gen  1 OOOOO--------- 
gen  2 O-O-O-O-O----- 
gen  3 O--O--O--O--O- = Euclid
gen  4 O-O-O---O---O- 
gen  5 OO---OO---O--- 
gen  6 O---O-O---O-O- 

Steps 14 events 6
Euclid
       --O-O-O--O-O-O
Gap
gen  1 OOOOOO-------- 
gen  2 O-O-O-O-O-O--- 
gen  3 OO-O--O--O--O- 
gen  4 O-O-O-O-O---O- 
gen  5 OO---OO---OO-- 
gen  6 O-O-O-O---O-O- 

Steps 14 events 7
Euclid
       -O-O-O-O-O-O-O
Gap
gen  1 OOOOOOO------- 
gen  2 O-O-O-O-O-O-O- = Euclid
gen  3 OO-OO-O--O--O- 
gen  4 O-O-O-O-O-O-O- = Euclid
gen  5 OOO--OO---OO-- 
gen  6 O-O-O-O-O-O-O- = Euclid

Steps 14 events 8
Euclid
       -O-O-OO-O-O-OO
Gap
gen  1 OOOOOOOO------ 
gen  3 OO-OO-OO-O--O- 
gen  5 OOO--OOO--OO-- 

Steps 14 events 9
Euclid
       -O-OO-OO-OO-OO
Gap
gen  1 OOOOOOOOO----- 
gen  3 OO-OO-OO-OO-O- = Euclid
gen  5 OOO--OOO--OOO- 

Steps 14 events 10
Euclid
       -OO-OOO-OO-OOO
Gap
gen  1 OOOOOOOOOO---- 
gen  3 OO-OO-OO-OO-OO 
gen  5 OOOO-OOO--OOO- 

Steps 14 events 11
Euclid
       -OOO-OOOO-OOOO
Gap
gen  1 OOOOOOOOOOO--- 
gen  3 OOOOO-OO-OO-OO 
gen  5 OOOO-OOOO-OOO- = Euclid

Steps 15 events 3
Euclid
       ----O----O----O
Gap
gen  1 OOO------------ 
gen  2 O-O-O---------- 
gen  3 O--O--O-------- 
gen  4 O---O---O------ 
gen  5 O----O----O---- = Euclid
gen  6 O-----O-----O-- 
gen  7 O------O------O 

Steps 15 events 4
Euclid
       ---O---O---O--O
Gap
gen  1 OOOO----------- 
gen  2 O-O-O-O-------- 
gen  3 O--O--O--O----- 
gen  4 O---O---O---O-- = Euclid
gen  6 O--O--O-----O-- 
gen  7 O-----OO------O 

Steps 15 events 5
Euclid
       --O--O--O--O--O
Gap
gen  1 OOOOO---------- 
gen  2 O-O-O-O-O------ 
gen  3 O--O--O--O--O-- = Euclid
gen  4 OO--O---O---O-- 
gen  6 O--O--O--O--O-- = Euclid
gen  7 O-----OO-----OO 

Steps 15 events 6
Euclid
       --O-O--O-O--O-O
Gap
gen  1 OOOOOO--------- 
gen  2 O-O-O-O-O-O---- 
gen  4 OO--OO--O---O-- 
gen  7 O----OOO-----OO 

Steps 15 events 7
Euclid
       --O-O-O-O-O-O-O
Gap
gen  1 OOOOOOO-------- 
gen  2 O-O-O-O-O-O-O-- = Euclid
gen  4 OO--OO--OO--O-- 
gen  7 O----OOO----OOO 

Steps 15 events 8
Euclid
       -O-O-O-O-O-O-OO
Gap
gen  1 OOOOOOOO------- 
gen  2 O-O-O-O-O-O-O-O = Euclid
gen  4 OO--OO--OO--OO- 
gen  7 O---OOOO----OOO 

Steps 15 events 9
Euclid
       -O-OO-O-OO-O-OO
Gap
gen  1 OOOOOOOOO------ 
gen  2 OOO-O-O-O-O-O-O 
gen  4 OOO-OO--OO--OO- 
gen  7 O---OOOO---OOOO 

Steps 15 events 10
Euclid
       -OO-OO-OO-OO-OO
Gap
gen  1 OOOOOOOOOO----- 
gen  2 OOOOO-O-O-O-O-O 
gen  4 OOO-OOO-OO--OO- 
gen  7 O--OOOOO---OOOO 

Steps 15 events 11
Euclid
       -OO-OOO-OOO-OOO
Gap
gen  1 OOOOOOOOOOO---- 
gen  2 OOOOOOO-O-O-O-O 
gen  4 OOO-OOO-OOO-OO- = Euclid
gen  7 O--OOOOO--OOOOO 

Steps 15 events 12
Euclid
       -OOOO-OOOO-OOOO
Gap
gen  1 OOOOOOOOOOOO--- 
gen  2 OOOOOOOOO-O-O-O 
gen  4 OOO-OOO-OOO-OOO 
gen  7 O-OOOOOO--OOOOO 

Steps 16 events 3
Euclid
       -----O----O----O
Gap
gen  1 OOO------------- 
gen  2 O-O-O----------- 
gen  3 O--O--O--------- 
gen  4 O---O---O------- 
gen  5 O----O----O----- = Euclid
gen  6 O-----O-----O--- 
gen  7 O------O------O- 

Steps 16 events 4
Euclid
       ---O---O---O---O
Gap
gen  1 OOOO------------ 
gen  2 O-O-O-O--------- 
gen  3 O--O--O--O------ 
gen  4 O---O---O---O--- = Euclid
gen  5 O----O----O----O 
gen  6 O-O---O-----O--- 
gen  7 O----O-O------O- 

Steps 16 events 5
Euclid
       ---O--O--O--O--O
Gap
gen  1 OOOOO----------- 
gen  2 O-O-O-O-O------- 
gen  3 O--O--O--O--O--- = Euclid
gen  5 O---OO----O----O 
gen  6 O-O---O-O---O--- 
gen  7 O----O-O----O-O- 

Steps 16 events 6
Euclid
       --O--O-O--O--O-O
Gap
gen  1 OOOOOO---------- 
gen  2 O-O-O-O-O-O----- 
gen  3 O--O--O--O--O--O 
gen  5 O---OO---OO----O 
gen  6 O-O---O-O---O-O- 
gen  7 O--O-O-O----O-O- 

Steps 16 events 7
Euclid
       --O-O-O--O-O-O-O
Gap
gen  1 OOOOOOO--------- 
gen  2 O-O-O-O-O-O-O--- 
gen  3 O-OO--O--O--O--O 
gen  5 O---OO---OO---OO 
gen  6 O-O-O-O-O---O-O- 
gen  7 O--O-O-O--O-O-O- = Euclid

Steps 16 events 8
Euclid
       -O-O-O-O-O-O-O-O
Gap
gen  1 OOOOOOOO-------- 
gen  2 O-O-O-O-O-O-O-O- = Euclid
gen  3 O-OO-OO--O--O--O 
gen  5 O--OOO---OO---OO 
gen  6 O-O-O-O-O-O-O-O- = Euclid
gen  7 OO-O-O-O--O-O-O- 

Steps 16 events 9
Euclid
       -O-O-O-OO-O-O-OO
Gap
gen  1 OOOOOOOOO------- 
gen  3 O-OO-OO-OO--O--O 
gen  5 O--OOO--OOO---OO 
gen  7 OO-O-O-OO-O-O-O- = Euclid

Steps 16 events 10
Euclid
       -O-OO-OO-O-OO-OO
Gap
gen  1 OOOOOOOOOO------ 
gen  3 O-OO-OO-OO-OO--O 
gen  5 O--OOO--OOO--OOO 
gen  7 OO-O-O-OO-O-O-OO 

Steps 16 events 11
Euclid
       -OO-OO-OO-OO-OOO
Gap
gen  1 OOOOOOOOOOO----- 
gen  3 O-OO-OO-OO-OO-OO = Euclid
gen  5 O-OOOO--OOO--OOO 
gen  7 OO-O-OOOO-O-O-OO 

Steps 16 events 12
Euclid
       -OOO-OOO-OOO-OOO
Gap
gen  1 OOOOOOOOOOOO---- 
gen  3 OOOO-OO-OO-OO-OO 
gen  5 O-OOOO-OOOO--OOO 
gen  7 OO-O-OOOO-O-OOOO 

Steps 16 events 13
Euclid
       -OOOO-OOOO-OOOOO
Gap
gen  1 OOOOOOOOOOOOO--- 
gen  3 OOOOOOO-OO-OO-OO 
gen  5 O-OOOO-OOOO-OOOO = Euclid
gen  7 OO-OOOOOO-O-OOOO

One thought on “Gaps in the sequence, part 2”

Pingback: G.E.A.R. SEQUENCER – Analog Output

Leave a comment Cancel reply

Published by Rich Holmes

#synthdiy #clarinet #bassclarinet #mcu #walking #bicycle #ukulele #sciencefiction #science #physics #math #space #rockets #genealogy #cis #male #straight #he #him #liberal #retired View all posts by Rich Holmes