# [seqfan] Some more new sequences

Robert Munafo mrob27 at gmail.com
Sat Jan 16 20:28:11 CET 2010

```During the hiatus I am using A-numbers I reserved from the "dispenser".

*"Hyper4" or Iterated Exponential Functions
*
There are two ways to define a^^b, distinguished by how you associate the
terms: a^^3 can be either (a^a)^a or a^(a^a). The first produces lower
values for b>2 and can be generalized easily to real or complex arguments.
The second produces higher values, is very hard to generalize to the reals,
but is more commonly accepted as the "correct" iterated exponential because
it cannot be expressed as a simple combination of other functions. The
following two sequences are ordered the same was as A003056 (addition),
A004247 (multiplication) and A003992 (exponentiation):

A171881: *0, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 27, 16, 1, 1, 5, 256, 19683, 256,
1, 1, 6, 3125, 4294967296, 7625597484987, 65536, 1, 1, 7, 46656,
298023223876953125, 340282366920938463463374607431768211456,
443426488243037769948249630619149892803, 4294967296, 1, 1, 8, 823543, ...*
Lower-valued (left-associative) version (square array read by
antidiagonals)

A171882: *1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 3, 4, 1, 1, 1, 4, 27, 16, 1, 0,
1, 5, 256, 7625597484987, 65536, 1, 1, 1, 6, 3125,
13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096,
...*
Higher-valued (right-associative) version (square array read by
antidiagonals)

*Accelerating Growth Sequences*

Back in the 1990's when I submitted A006888, I had dozens more similar
sequences in the margins of my Handbook of Integer Sequences. They are all
an effort to create a recurrence-generated sequence that "starts out slow"
and then grows faster and faster.

A171874 : *0, 0, 0, 1, 1, 2, 4, 7, 16, 46, 174, 3311, 268446771,
401906756202069927727330981, ...*
A171877 : *0, 0, 1, 1, 1, 3, 5, 9, 25, 73, 423, 61297, 3814697357801,
38288777744833624093154249190851262684887027, ...
*A171878 : *0, 0, 0, 0, 1, 2, 3, 6, 13, 33, 120, 765, 4831534,
55040353993453427047,
410186270246002225336426103593500672000000000000055040353997149550557, ...*
Each of these uses the recurrence
A[N]=A[N-1]+A[N-2]*A[N-3]+A[N-4]^A[N-5], after the initial 5 terms. Note
that 0^0=1.

A171879 : *0, 0, 1, 1, 1, 1, 3, 5, 9, 25, 73, 313, 3263, 1502337,
278472902914281, 11984387434132924341157279996736444304839056033321, ...*
A171880 : *0, 0, 0, 1, 1, 1, 2, 4, 7, 16, 46, 166, 1014, 47066,
12348246366,66716521529543607970475115226, ...*
Both of these use the recurrence
A[N]=A[N-1]+A[N-2]*A[N-3]+A[N-4]*A[N-5]^A[N-6], after the initial 6 terms.
Again, note that 0^0=1.

*From Link Between Classification-Counting Sequence A171871 and Binary Block
Codes A039754*

A171876 : *1, 1, 1, 1, 1, 3, 3, 1, 1, 4, 6, 19, 27, 50, 56, 1, 1, 5, 10, 47,
131, 472, 1326, 3779, 9013, 19963, 38073, 65664, 98804, 133576, 158658, 1,
1, 6, 16, 103, 497, 3253, 19735, 120843, 681474, 3561696, ...*
These are the terms found in both A039754 and the "Franklin's triangle"
that came up from Andrew Weimholt's and my work on A005646. The latter uses
another reserved number (A171871) but not yet in OEIS.

A171875 : *0, 0, 1, 3, 17, 74, 358, 1631, 7563, 34751, 160807, ...*
Subdiagonal of triangle A171871. The last term results from my recent
extension of A005646 to 12 terms.

All of these have more detail at:
http://mrob.com/pub/math/OEIS-extra.txt
http://mrob.com/pub/math/hyper4.html
http://mrob.com/pub/math/seq-accelerate.html
http://mrob.com/pub/math/seq-a005646.html

--
Robert Munafo  --  mrob.com

```