OFFSET
1,2
COMMENTS
Conjecture: where records occur in A237271. - Omar E. Pol, Dec 27 2016
This sequence of (first occurrence of) parts appears to be strictly increasing in contrast to sequence A250070 of (first occurrence of) maximum widths. - Hartmut F. W. Hoft, Dec 09 2014
It appears that all terms are odd numbers. - Omar E. Pol, Oct 14 2018
Let n = 2^m * q with m>0 and q odd; then the 1's in even positions of row n in the triangle of A237048 are at positions 2^(m+1) * d <= row(n) where d divides q. For n/2 the even positions of 1's occur at the smaller values 2^m * d <= row(n/2), thus either keeping or reducing widths (A249223) of parts in the symmetric representation of sigma for n/2 inherited from row n. Therefore the number of parts for n is at most as large as for n/2, i.e., all numbers in this sequence are odd. - Hartmut F. W. Hoft, Sep 22 2021
Observation: at least for n = 1..21 we have that 2*a(n) < a(n+1). - Omar E. Pol, Sep 22 2021
LINKS
Hartmut F. W. Hoft, Procedural implementation for extension values
Omar E. Pol, Perspective view of the stepped pyramid (16 levels)
EXAMPLE
------------------------------------------------------
------------------------------------------------------
1 1 [1] 1
2 3 [2, 2] 4
3 9 [5, 3, 5] 13
4 21 [11, 5, 5, 11] 32
5 63 [32, 12, 16, 12, 32] 104
...
For n = 3 the symmetric representation of sigma(9) = 13 contains three parts [5, 3, 5] as shown below:
.
. _ _ _ _ _ 5
. |_ _ _ _ _|
. |_ _ 3
. |_ |
. |_|_ _ 5
. | |
. | |
. | |
. | |
. |_|
.
MATHEMATICA
(* a239663[] permits computation in intervals *)
(* Function a237270[] is defined in A237270 *)
(* variable "list" contains the first occurrences up to m *)
a239663[list_, {m_, n_}]:=Module[{firsts=list, g=Length[list], i, p}, For[i=m, i<=n, i++, p=Length[a237270[i]]; If[p>g, AppendTo[firsts, i]; g=p]]; firsts]
a239663[{1}, {1, 1000}] (* computes the first 8 values *)
(* Hartmut F. W. Hoft, Jul 08 2014 *)
a239663[n_, len_] := Module[{list=Table[0, len], i, v}, For[i=1, i<=n, i+=2, v=Count[a341969[i], 0]+1; If[list[[v]]==0, list[[v]]=i]]; list]
a239663[62000000, 22] (* Hartmut F. W. Hoft, Sep 22 2021 *)
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Omar E. Pol, Mar 23 2014
EXTENSIONS
a(6)-a(8) from Michel Marcus, Mar 28 2014
a(9) from Michel Marcus, Mar 29 2014
a(10)-a(11) from Michel Marcus, Apr 02 2014
a(12) from Hartmut F. W. Hoft, Jul 08 2014
a(13)-a(18) from Hartmut F. W. Hoft, Dec 09 2014
a(19)-a(22) from Hartmut F. W. Hoft, Sep 22 2021
STATUS
approved