A061762 a(n) = (sum of digits of n) + (product of digits of n).
N. J. A. Sloane
njas at research.att.com
Wed Aug 22 13:33:38 CEST 2007
On 8/20/07 Max wrote :
---------
On 8/9/07, koh <zbi74583 at boat.zero.ad.jp> wrote:
> Once I considered about "Strokes" of a graph.
>
> See the definition in A089243.
>
> Or, "Partition of a graph G into strokes S_i" must satisfy the following conditions.
>
> o Union_{i} S_i = H
> o If not{i=j} -> S_i and S_j don't have the same edge
> o If not{i=j} -> S_i U S_j isn't a dipath
> o For all i S_i is a dipath
> Where H is a digraph on G
[...]
> %S A000001 1,2,4,8,16,32,64
> %N A000001 Number of partitions of G_n into "strokes".
> G_n = {V_n, E_n}, V_n = {v_1, v_2,….v_n}, E_n = {v_1v_2, v2_v_3,….v_{n-1}v_n}
>
> Figure of G_5 : o-o-o-o-o
This particular sequence (which is in OEIS under the name A131517)
seems to be a duplicate of A000079...
I cannot say for sure until I understand your definition of "partition
into strokes".
For now I have a couple of questions:
1) Shouldn't it be a directed graph like
G_5: o -> o -> o -> o -> o
as the partition into strokes is defined only for directed graphs?
2) Could you list all partitions, say, for n=3 ?
Thanks,
Max
----------
I have not submitted these sequences.
I wander Neil did it.
To Neil :
It was too fast.
I am not sure that some of these are exact number.
I must calculate them more.
To Max :
I think that the two sequences are the same, but I have no proof for it.
Your responce make me think a little, and after five minutes I have got a proof.
It is the same as A000079.
And A131520 must be :
%S A131520 2,6,12,22,40,74,140
%F A131520 a(n)=2^n+2*n-2
1) You are right.
But I wrote a new definition for a graph.
"Partition of a graph G into strokes" means "Partition of a digraph H on graph G into strokes".
See the four conditions.
2) n=3
o-o-o names of vertices 1-2-3
Partitions into strokes :
1->2->3
3->2->1
1->2, 3->2
2->1, 2->3
So, a(3)=4
Yasutoshi
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