# children of safe pilings

vdmcc w.meeussen.vdmcc at vandemoortele.be
Tue Jun 15 18:23:01 CEST 1999

```hi seqfanners,

the sequences below are probably too short & too farfetched to be of genuine
&
general interest. However, it might be worth considering if there are many
non-monotonous sequences that are a union of three monotonous subsequences
as in the cases below. It stands to reason that the safe 3D-pilings must
show
a modularity of 3. It is even to be expected that there should be a
recursion
for each subsequence. If I where intelligent enough, I'd find them.
If Neil chooses to include them in EIS, what description should be used ???

table of {n, number of C3 safe pilings of n boxes}
{{13,2},{14,2},{16,2},{17,2},{19,4},{20,4},{22,6},{23,6},{25,10},{26,12},{28
,
14},{29,18},{31,22},{32,32},{33,2},{34,28},{35,46},{36,4},{37,40},{38,

72},{39,8},{40,54},{41,104},{42,14},{43,74},{44,156},{45,26},{46,96},{47,
222},{48,42},{49,130},{50,326},{51,72},{52,166},{53,454},{54,112},{55,

218},{56,644},{57,178},{58,278},{59,888},{60,270},{61,358},{62,1236},{63,
414},{64,452}}

number of safe pilings with C3-symmetry  with 3n-2 elements (Offset=1)
{0,0,0,0,1,1,2,3,5,7,11,14,20,27,37,48,65,83,109,139,179,226}
status in EIS : very close match with A035961
but probably not the same (* see A035961 at end of this mail *)

number of C3 safe pilings with 3n-1 elements (Offset=1)
{0,0,0,0,1,1,2,3,6,9,16,23,36,52,78,111,163,227,322,444,618}
I am sorry, but the terms
1,2,3,6,9,16,23,36
do not match anything in the table

number of C3 safe pilings with 3n   elements (Offset=1)
{0,0,0,0,0,0,0,0,0,0,1,2,4,7,13,21,36,56,89,135,207}
I am sorry, but the terms
1,2,4,7,13,21,36,56
do not match anything in the table
-------------------------------------------------------
table of {n, number of C3v safe pilings}
{{1,1},{4,1},{7,2},{8,1},{10,2},{11,1},{13,2},{14,1},{16,3},{17,2},{19,4},{2
0,
4},{22,4},{23,5},{25,5},{26,7},{27,1},{28,6},{29,9},{30,1},{31,6},{32,

11},{33,1},{34,8},{35,15},{36,2},{37,10},{38,20},{39,3},{40,10},{41,25},{
42,4},{43,12},{44,33},{45,7},{46,14},{47,40},{48,9},{49,15},{50,48},{51,
12},{52,18},{53,60},{54,17},{55,20},{56,74},{57,23},{58,22},{59,89},{60,
30},{61,26},{62,108},{63,40},{64,30}}

number of C3v safe pilings with 3n-2 elements (Offset=1)
{1,1,2,2,2,3,4,4,5,6,6,8,10,10,12,14,15,18,20,22,26,30}
I am sorry, but the terms
1,2,2,2,3,4,4,5,6,6,8,10
do not match anything in the table

number of C3v safe pilings with 3n-1 elements (Offset=1)
{0,0,1,1,1,2,4,5,7,9,11,15,20,25,33,40,48,60,74,89,108}
I am sorry, but the terms
1,2,4,5,7,9,11,15
do not match anything in the table

number of C3v safe pilings with 3n elements (Offset=1)
{0,0,0,0,0,0,0,0,1,1,1,2,3,4,7,9,12,17,23,30,40}
I am sorry, but the terms
1,2,3,4,7,9,12
do not match anything in the table
---------------------------------------------------------
(*
last place=226 differs from a 'close' sequence :
ID Number: A035961
Sequence:
1,2,3,5,7,11,14,20,27,37,48,65,83,109,139,179,225,287,357,449,556,691,

848,1047,1276,1561,1893,2299,2772,3348,4015,4820,5756,6874,8171,9716,
11501,13614,16058,18932,22249,26138,30613,35838,41848,48831
Name:      Partitions in parts not of the kind 15k, 15k+7 or 15k-7. Also
partitions with at most 6
parts of size 1 and differences between parts at distance 6 are
greater than 1.
Comments:  Case k=7,i=7 of Gordon Theorem.
References G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p.
109.
Keywords:  nonn,easy,partOffset:    1
Author(s): Olivier Gerard (ogerard at ext.jussieu.fr)

Olivier sent me the %t-line :
GF = 1/Product[(1-Switch[Mod[k,15],0,0,7,0,8,0,_,x^k]),{k,1,oo}]
no chance of a one-to-one correspondence that I can see. He neither.
*)

[[irony : try to extend some sequences to reasonable length,
and you breed twice as much ugly short onces]]

w.meeussen.vdmcc at vandemoortele.be
tel  +32 (0) 51 33 21 11
fax +32 (0) 51 33 21 75

```