What is "axially"?

[Joshua Zucker]

>  >  A006746                  Axially symmetric polyominoes with n cells.

Polyominoes with n cells and exactly one line of reflection symmetry,
where that one line is parallel to the grid.

>  >  A030227                  Number of n-celled polyominoes with bilateral symmetry.

Polyominoes with n cells and at least one line of reflection symmetry.

--Joshua




These were suggested by the contents page of a book,
Yuan Wang, editor, The Goldbach Conjecture,

a(n) = number of ways to write 2n+1 as the sum of three primes

a(n) = number of ways to write 2n as the sum of a prime and an

a(n) = number of ways to write 2n as the sum of two almost primes.

In fact they may already be in the OEIS - could someone please check?
(There could be several versions of the first and third ones.)

Thanks

Neil




njas> From seqfan-owner at ext.jussieu.fr  Mon Mar 10 10:08:02 2008
njas> Date: Sun, 9 Mar 2008 23:43:49 -0400
njas> From: "N. J. A. Sloane" <njas at research.att.com>
njas> To: seqfan at ext.jussieu.fr
njas> Subject: 3 potentially new sequences
njas> Cc: njas at research.att.com
njas> ....
njas> a(n) = number of ways to write 2n+1 as the sum of three primes
njas> 
njas> a(n) = number of ways to write 2n as the sum of a prime and an
njas>   almost prime
njas> 
njas> a(n) = number of ways to write 2n as the sum of two almost primes.
njas> 
njas> In fact they may already be in the OEIS - could someone please check?
njas> (There could be several versions of the first and third ones.)
njas> ...

Some variants of the first a(n) with A-numbers where I could identify
them (independent analysis encouraged!). 2n+1 is a bi-section of
an associated "write n as the sum.." sequence:

number of ways writing 2n+1 as the sum of 1, 2 or 3 primes, n=1,2..
1, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 8, 8, 10, 8, 10, 11, 12, 12, 13, 14, 13,
15

number of ways writing 2n+1 as the sum of exactly 3 primes, n=1,2.. A054860?
0, 0, 1, 2, 2, 2, 3, 4, 3, 5, 5, 5, 7, 7, 6, 9, 8, 9, 10, 11, 10, 12, 13, 12, 
15

number of ways writing n as the sum of 1, 2 or 3 primes, n=1,2.. A071335?
0, 1, 1, 1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 5, 4, 5, 4, 6, 5, 6, 6, 6

number of ways writing n as the sum of exactly 3 primes, n=1,2.. A068307?
0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 4, 2, 3, 2, 5, 2, 5, 3, 5

number of ways writing 2n+1 as the sum of 1, 2 or 3 distinct primes, n=1,2..
1, 2, 2, 1, 1, 2, 2, 1, 3, 3, 3, 3, 3, 4, 6, 5, 5, 6, 7, 7, 8, 7, 10, 9, 8

number of ways writing 2n+1 as the sum of exactly 3 distinct primes, n=1,2.. A102605?
0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 9, 8, 8

number of ways writing n as the sum of 1, 2 or 3 distinct primes, n=1,2..
0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 4, 3, 4, 3, 4, 3, 5, 3

number of ways writing n as the sum of exactly 3 distinct primes, n=1,2..A125688?
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 2, 2, 2, 2

number of ways writing 2n+1 as the sum of exactly 2 primes, n=1,2..
0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0

number of ways writing n as the sum of exactly 2 primes, n=1,2.., A061358?
0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 0, 2, 1, 2, 1, 3, 0, 3, 1

number of ways writing 2n+1 as the sum of exactly 2 distinct primes, n=1,2..
0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0

number of ways writing n as the sum of exactly 2 distinct primes, n=1,2.. A117929?
0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 2, 0, 3, 1


# (ugly!) Maple source code for those who need more terms in a hurry:
interface(prettyprint=0) :
interface(quiet=true) :
# exactly two primes, both >= pmin (must be distinct if dist=true)
a2pr := proc(n,pmin,dist)
end:

# n into sum of three primes count
# exactly 3 if exCnt =true, must be distinct if dist=true
a3pr := proc(n,exCnt,dist)
end:

nmax := 25:
printf("\nnumber of ways writing 2n+1 as the sum of 1, 2 or 3 primes, n=1,2..\n") ;
printf("\nnumber of ways writing 2n+1 as the sum of exactly 3 primes, n=1,2.. A054860?\n") ;
printf("\nnumber of ways writing n as the sum of 1, 2 or 3 primes, n=1,2.. A071335?\n") ;
printf("\nnumber of ways writing n as the sum of exactly 3 primes, n=1,2.. A068307?\n") ;

printf("\nnumber of ways writing 2n+1 as the sum of 1, 2 or 3 distinct primes, n=1,2..\n") ;
printf("\nnumber of ways writing 2n+1 as the sum of exactly 3 distinct primes, n=1,2.. A102605?\n") ;
printf("\nnumber of ways writing n as the sum of 1, 2 or 3 distinct primes, n=1,2..\n") ;
printf("\nnumber of ways writing n as the sum of exactly 3 distinct primes, n=1,2..A125688?\n") ;


printf("\nnumber of ways writing 2n+1 as the sum of exactly 2 primes, n=1,2..\n") ;
printf("\nnumber of ways writing n as the sum of exactly 2 primes, n=1,2.., A061358?\n") ;
printf("\nnumber of ways writing 2n+1 as the sum of exactly 2 distinct primes, n=1,2..\n") ;
printf("\nnumber of ways writing n as the sum of exactly 2 distinct primes, n=1,2.. A117929?\n") ;