[seqfan] Sequence (a triangle) related to A141419, not in OEIS?

Ed Jeffery ed.jeffery at yahoo.com
Tue Nov 22 03:04:49 CET 2011


Seqfans,

I have a sequence (actually several) related to A141419:

https://oeis.org/A141419

We have 


A141419={1,2,3,3,5,6,4,7,9,10,5,9,12,14,15,...},


which in triangular form is

{1}, 
{2, 3}, 
{3, 5, 6}, 
{4, 7, 9, 10}, 
{5, 9, 12, 14, 15}, 
{6, 11, 15, 18, 20, 21}, 
{7, 13, 18, 22, 25, 27, 28}, 
{8, 15, 21, 26, 30, 33, 35, 36}, 
{9, 17, 24, 30, 35, 39, 42, 44, 45}, 
{10, 19, 27, 34, 40, 45, 49, 52, 54, 55}
etc.

If you read my recent comment for 
this sequence, then you'll see that the rows of the triangle are identical to the first rows of a certain sequence of matrices. 


Similarly, a (new) sequence I propose is found from the second rows of that same 
sequence of matrices, with the n-th matrix (for n>1) being related to a certain 
class of rhombus substitution tilings showing (2*n+1)-fold rotational symmetry. For n=1,2,3,..., and k=0,1,...,n-1, the proposed sequence in triangular form, setting t(1,0)=1, is:

t(n,k)=

{1};
{3, 5};
{5, 9, 11};
{7, 13, 17, 19};
{9, 17, 23, 27, 29};
{11, 21, 29, 35, 39, 41};
{13, 25, 35, 43, 49, 53, 55};...

with the first three diagonals evidently essentially

{1,5,11,19.29,41,55,...}=A028387 (https://oeis.org/A028387),
{3,9,17,27,39,53,69,...}=A014209(m+1), m=0,1,2,... (https://oeis.org/A014209) 

and

{5,13,23,35,49,65,83,...}=A108195 (https://oeis.org/A108195).

I don't have a closed-form formula for this sequence yet, butdefining t(n,0)=2*n-1, we have, for n>1, the obvious relation between successive rows,

t(n,k)=t(n,0)+t(n-1,k-1)+1   (0<k<n).


I can't find any of the other diagonals as sequences in OEIS. But for given n in {1,2,3,...}, the diagonal sequence beginning with the triangle entry t(n,0) and defined as 


a_n={a_n(r)}={t(n+r,r}}, r=0,1,2,..., 


we have that the n-th such sequence (or diagonal of the triangle) is (evidently) given by


a_n(r)=r^2+(2*n-1)*r-1, 


assuming I haven't messed up the calculations.

Is this sequence interesting enough to include, and could someone else please verify that it isn't in the database yet? I already spent a lot of time looking, so your help will be greatly appreciated.

Thanks and regards,

Ed Jeffery


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