# [seqfan] surreal numbers and A025480

Jeff Anderson peawormsworth at gmail.com
Thu Nov 19 01:05:06 CET 2020

```Question: Is this observation worth of entry into your database?
Question: Should I say: "This sequence is the concatenation of all
sequences of surreal number indexes ordered by value for consecutive days
starting from zero."

==============
Consider sequence A025480:

0
0                                               1
0                       2                       1                       3
0           4           2           5           1           6           3
7
0     8     4     9     2    10     5    11     1    12     6    13     3
14     7    15
0 16  8 17  4 18  9 19  2 20 10 21  5 22 11 23  1 24 12 25  6 26 13 27  3
28 14 29  7 30 15 31

...as an index of birth order for Surreal number values:

N
N                                               0
N                      -1                       0                       1
N          -2          -1        -1/2           0         1/2           1
2
N    -3    -2   -3/2   -1  -3/4  -1/2  -1/4     0   1/4   1/2   3/4     1
3/2     2     3

N -4 -3 -5 -2 -7 -3 -5 -1 -7 -3 -5 -1 -3 -1 -1  0  1  1  3  1  5  3  7  1
9  3 11  2  5  3
1  1  1  2  1  4  2  4  1  8  4  8  2  8  4  8  1  8  4  8  2  8  4  8  1
8  2  8  1  2  3  4

* last two lines are a single entry having a numerator (top) and
denominator (bottom)
* N = Nan or Void. Considered less than all numbers here.

... The original sequence is the concatenation of all sequences of surreal
number indexes ordered by value for consecutive days starting from zero.

Perhaps this is more clear...

For surreal Numbers:

Day  Order of Values                 Index sorted by Value
0    {N}                             {0}
1    {N,0}                           {0,1}
2    {N,0,-1,1}                      {0,2,1,3}
3    {N,0,-1,1,-2,-1/2,1/2,2}        {0,4,2,5,1,6,3,7}
4    {N,0,-1,1,-2,-1/2,1/2,2,-3,-5/2,-3/4,-1/4,1/4,3/4,5/2,3}
{0,8,4,9,2,10,5,11,1,12,6,13,3,14,7,15}
5    ...etc...

Where A025480 are the concatenation of these value sorted index set
elements.

Jeff

```