[seqfan] Re: sequence defined by ``exotic addition''

[franktaw at netscape.net]

The first component of n x [1,1] is n.  Call the second a(n).  Then we 
have:

a(2n) = 2n a(n).
a(2n+1) = 2n  + a(2n) = 2n (a(n) + 1)

This definition is similar to A098844.

I get values that agree with Maximillian's, not those in the original 
message:

1, 2, 4, 8, 12, 24, 30, 64, 72, 120, 130, 288, 300, 420, 434, 1024, 
1040, 1296, 1314, 2400, 2420, 2860, 2882, 6912, 6936, 7800, 7826, 
11760, 11788, 13020, 13050, 32768, 32800, 35360, 35394, 46656, 46692, 
49932, 49970, 96000

-----
It is unlikely that an associative operation would give a more 
interesting sequence; more likely it would produce a very simple 
sequence.

-----
You have to take some of superseeker's responses with a grain of salt.  
If you send in 16 terms, and it finds a degree 15 g.f., you haven't 
learned anything.

With the revised sequence, superseeker suggests

[64 - 192 a(n) + 240 a(n)^2  - 160 a(n)^3  + 60 a(n)^4  - 12 a(n)^5  + 
a(n)^6 , lgdegf]

(lgdegf  =   logarithmic derivative of exponential generating function)

I sent in 18 terms, so this may well be real.

Franklin T. Adams-Watters

-----Original Message-----
From: Georgi Guninski <guninski at guninski.com>

define binary operation "o" on pairs a,b:
a o b = a[1]+b[1] , a[1]*b[2]+a[2]*b[1]

define scalar multiplication "x":
2n x a = (n x a) o (n x a)
2n+1 x a= ((n x a) o (n x a)) o a
1 x a = a


the sequence a(n) is the second component of n x [1,1]:
1,2,4,8,12,24,40,64,72,144,224,320,416,576,768,1024,1040 ...

the sequence may be more interesting if "o" were associative, is there a
list of binary associative functions?

attached is a sample pari implementation.

superseeker claims to find degree 15 revogf.

thanks.

--
georgi