Hello,

This sequence S is equal to its fourth |absolute differences|:

Seq. S :†† 7153 14 2 10 6 28 4 20 12†† 56 8†† 40 24 112 16 80 48224†† 32†† 160 96448 ...

First dif. 6 ††4 ††2 1112 8 ††4 22 24 16†† 8 ††44†† 4832 16 88 ††96 6432 176†† 19212864 352 ...

Second dif.††2 ††2 ††9††1 ††44 18 2 ††8†† 8 364 ††16 16 72 ††832†† 32 144††† 16†† 64††† 64288 ...

Third diff.†† †††0 ††78†††3014 16 6†† 0 2832 12†† 0††5664†† 24†† 0 112†† 128††† 48†† 0†† 224 ...

Fourth diff.†† ††††71 ††5††3 ††14210628†† 4††20†† 12 568†† 40†† 24 112††† 16††† 80††† 48224 ... = 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) doesnít loop on itself -- but enters a different 4th-differences-loop sequence:

 

Seq. S: ††††1357 2 6 10 14 4 12 20 28†† 8 24†† 40 56 16 48 80 1123296†† 160 22464 ...

|First dif.| 2 ††2 ††2 544 ††4 10 8 ††8†† 8 ††20†† 1616 16 40 32 3232 80†† 64†† 64 64 160 ...

|Second dif.|††0 ††0 ††31 ††00 ††6 2 00 124 ††††0 0 24 80†† 0 48††† 160††† 096 ...

|Third diff.|†† †††0 ††3210 6 4 20 128 4††02416†† 80 48††32††† 16†† 0†† 96 ...

|Fourth diff.|†† ††††31 ††11 ††622†† 2††12†† 4††† 4†† ††4 ††24†† 8††† 8††† 8†† 4816††† 16††† 1696

|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†† 160†† 408††† 32††† 0†††† 80 ...

|Seventh differences|††††† 2††† 5†† 4††† 3††† 4†† 108††† 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 ††22 4 4 4 88 8 161616323232646464128128128 256 ...

00†† 1†† 0†† 0†† 2†† 0†† 0†† 4†† 0†† 0†† 8††† 0†† 0 16†† 0†† 032 0†† 064††† 0††† 0 ††128 ...

††† 0†† 1†† 1†† 0†† 2†† 2†† 0†† 4†† 4†† 0†† 8†† 8††† 016160†† 3232064†† 64††† 0128 ...

††††† 1†† 0†† 1†† 2†† 0†† 2†† 4†† 0†† 4†† 8†† 0††† 016†† 0†† 1632†† 03264 0††† 64128 ...

††††††† 1 1 1 2 ††22 4 4 4 88 8 1616 16323232646464 ...

 

 

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††† 3216†† 64†† 32128††† 64†† 256†† 128†† 512†† 256†† 1024†† 512††† 2048 ...

††† 1†† 3†† 2†† 6†† 4†† 12†† 8†† 24†† 16†† 48†† 32†† 9664†† 192†† 128†† 384†† 256†† 768†† 512†† 1536 ...

 

 

Those are self looping (differences neednít 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†† 5121024†† 2048†† 4096 ...

 

A007283:

3†† 6†† 12†† 24†† 48†† 96†† 192†† 384†† 768†† 1536†† 3072†† 6144†† 12288 ...

3†† 6†† 12†† 24†† 48†† 96†† 192†† 384†† 7681536†† 3072†† 6144 ...

 

 

Could someone find a 3-step looping sequence?

 

Is there a general formula to produce such sequences?

 

Is this old hat?

 

Best,

….