Hello,
This sequence S
is equal to its fourth |absolute differences|:
Seq. S : 7 1 5 3
14 2 10 6 28 4 20 12 56 8 40 24 112 16 80 48 224
32 160 96 448 ...
First dif. 6 4 2 11 12 8 4 22 24 16 8 44
48 32 16 88 96 64 32 176
192 128 64 352 ...
Second dif. 2 2 9 1 4 4 18 2 8 8 36 4 16 16 72 8 32 32 144
16 64 64 288 ...
Third diff. 0 7 8 3 0 14 16 6 0 28 32 12
0 56 64
24 0 112
128 48 0 224 ...
Fourth diff. 7 1 5 3 14 2 10 6 28 4 20 12
56
8 40 24 112
16 80 48
224 ... = Sequence S
S has this structure:
a b c d 2a 2b 2c 2d 4a 4b 4c 4d 8a 8b 8c
8d 16a 16b 16c 16d... where (a,b,c,d) are (7,1,5,3)
Note
that the sequence based on (a,b,c,d) = (1,3,5,7) doesn’t loop on itself -- but
enters a different 4th-differences-loop sequence:
Seq. S: 1 3 5 7 2 6 10 14 4 12 20 28 8 24 40 56 16 48 80 112 32 96 160
224 64 ...
|First dif.| 2 2 2 5 4 4 4 10 8 8 8 20 16 16 16 40 32 32 32 80
64 64 64 160 ...
|Second dif.| 0 0 3 1 0 0 6 2 0 0 12 4 0 0 24 8 0 0 48
16 0 0 96 ...
|Third diff.| 0 3 2 1 0 6 4 2 0 12 8 4 0 24 16 8 0 48 32 16 0 96 ...
|Fourth diff.|
3 1 1 1 6 2 2 2 12 4 4 4 24 8 8 8 48 16 16 16 96
|Fifth differences|
2 0 0 5
4 0 0 10
8 0 0 20
16 0 0 40 32
0 0 80
|Sixth differences| 2
0 5 1
4 0 10 2 8 0 20
4 16 0 40 8
32 0 80 ...
|Seventh differences| 2
5 4 3
4 10 8 6
8 20 16
12 16 40
32 24 32 80 ...
|Eighth differences|
3 1 1 1 6 2 2 2 12 4 4 4 24 8 8 8 48 ... = |Fourth
diff.|
This
sequence loops in 4 steps too:
1 1 1 2 2 2 4 4 4 8 8 8 16 16 16 32 32 32 64 64 64 128 128 128 256 ...
0 0 1
0 0 2
0 0 4
0 0 8 0 0 16 0 0 32 0
0 64
0 0 128 ...
0
1 1 0
2 2 0
4 4 0
8 8 0 16 16 0 32 32 0 64 64 0
128 ...
1
0 1 2 0 2
4 0 4
8 0 0 16 0 16 32
0 32 64 0 64
128 ...
1 1 1 2 2 2 4 4 4 8 8 8 16 16
16 32 32 32 64 64 64 ...
This
one loops in 2 steps:
1 3 2 6 4 12 8 24 16 48 32 96 64 192 128 384 256 768 512 1536
1024 3072 ...
2
1 4 2
8 4 12
8 32 16
64 32 128
64 256 128
512 256 1024
512 2048 ...
1 3 2 6 4 12 8 24 16 48 32 96 64 192 128 384 256 768 512 1536 ...
Those
are self looping (differences needn’t to be absolute, here):
A000079:
1 2 4 8 16 32 64 128 256 512 1024
2048 4096 8192 ...
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 ...
A007283:
3 6 12 24 48 96 192 384 768 1536
3072 6144 12288 ...
3 6 12 24 48 96 192 384 768 1536 3072 6144 ...
Could someone find a 3-step looping sequence?
Is there a general formula to produce such sequences?
Is this old hat?
Best,
É.