[seqfan] Some integer sequences being irrational iterates of another integer sequences

Thomas Baruchel baruchel at gmx.com
Sun May 10 09:51:06 CEST 2020

```Dear fellow seqfans,

I investigated some ideas related to fractional iterates of functions
(see http://dmishin.blogspot.com/2018/11/fractional-iterates-of-x-2-problem-by.html
for informations about that).

Referring here to generating functions (ogf) only when speaking about a
sequences, I looked for sequences in the database which could be built
as the k-th iterate of some other integer sequence, with k being some
positive irrational exponent, and I found some of them.

The challenging part was computing an irrational iterate of some power
series with integer coefficients and finding another power series with
integer coefficients. Here are some I gathered:

B = A^( log(3)/log(2) )
with A = [0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, ... ]
and  B = [0, 3, 0, 8, 0, 24, 0, 72, 0, 216, 0, 648, 0, 1944, ... ]

B = A^( log(3)/log(2) )
with A = [0, 2, -1, 4, -2, 8, -4, 16, -8, 32, -16, 64, -32, ... ]
and  B = [0, 3, -3, 17, -24, 108, -178, 696, -1284, 4532, -9096, ... ]

B = A^( log(3)/log(2) )
with A = [0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, ... ]
and  B = [0, 3, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, ... ]
B = A^( log(5)/log(2) )
with A = [0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, ... ]
and  B = (0 5 40 320 2560 20480 163840 1310720 10485760 83886080 671088640 5368709120
B = A^( log(6)/log(2) )
with A = [0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, ... ]
and  B = (0 6 60 600 6000 60000 600000 6000000 60000000 600000000 6000000000 60000000000
etc.

B = A^( log(3)/log(2) )
with A = (0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2)
and  B = (0 3 6 12 24 48 96 192 384 768 1536 3072 6144 12288 24576 49152 98304 196608
B = A^( log(5)/log(2) )
with A = (0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2)
and  B = (0 5 20 80 320 1280 5120 20480 81920 327680 1310720 5242880 20971520 83886080
B = A^( log(6)/log(2) )
with A = (0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2)
and  B = (0 6 30 150 750 3750 18750 93750 468750 2343750 11718750 58593750 292968750
etc.

B = A^( log(7)/log(4) )
with A = (0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4)
and  B = (0 7 14 28 56 112 224 448 896 1792 3584 7168 14336 28672 57344 114688 229376
B = A^( log(25)/log(4) ) = A^( log(5)/log(2) )
with A = (0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4)
and  B = (0 25 200 1600 12800 102400 819200 6553600 52428800 419430400 3355443200

B = A^( log(9)/log(4) ) = A^( log(3)/log(2) )
with A = (0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100
and  B = (0 9 48 208 816 3024 10800 37584 128304 431568 1434672 4723920 15431472
B = A^( log(25)/log(4) ) = A^( log(5)/log(2) )
with A = (0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100
and  B = (0 25 400 5360 65680 763376 8568720 93837040 1008877200 10693257200

B = A^( log(7)/log(4) )
with A = (0 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144
and  B = (0 7 28 112 448 1792 7168 28672 114688 458752 1835008 7340032 29360128
B = A^( log(25)/log(4) ) = A^( log(5)/log(2) )
with A = (0 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144
and  B = (0 25 400 6400 102400 1638400 26214400 419430400 6710886400 107374182400

B = A^( log(3)/log(2) )
with A = (0 2 6 18 54 162 486 1458 4374 13122 39366 118098 354294 1062882 3188646
and  B = (0 3 18 108 648 3888 23328 139968 839808 5038848 30233088 181398528 1088391168
B = A^( log(5)/log(2) )
with A = (0 2 6 18 54 162 486 1458 4374 13122 39366 118098 354294 1062882 3188646
and  B = (0 5 60 720 8640 103680 1244160 14929920 179159040 2149908480 25798901760
B = A^( log(6)/log(2) )
with A = (0 2 6 18 54 162 486 1458 4374 13122 39366 118098 354294 1062882 3188646
and  B = (0 6 90 1350 20250 303750 4556250 68343750 1025156250 15377343750 230660156250
etc.

Regards,

--
th. baruchel
```

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