# [seqfan] Triangular Numbers A000217, Linear Search Results

Fri Jul 17 09:42:36 CEST 2015

```Hey Seqfans,

This message shares the results of a search which considers sequences as
vectors with components a_n. A choice of basis restricts the search to a
three-dimensional vector space that contains the triangular numbers. The
g.f.-basis is:

vgf = {
gf( A079978 ) ,
gf( A105637 ) ,
gf( A069905 )
}

vgf = {
1/(1 - x^3),
x*(1 + 2*x)/((1 - x^2)*(1 - x^3)) ,
x^3/((1 - x)*(1 - x^2)*(1 - x^3))
}

In this g.f.-basis an integer 3-tuple specifies an integer sequence g.f. up
to some power of x that determines a left / right shift.

For example,

gf(A000217) =   x { 1, 3, 6 } * gf,

where "*" denotes inner product.

By searching and checking generating functions we obtain the following
results:

A230059, {0,1,1}, Zeta Conjecture?
A001840, {0,1,2}, Triangular Choosing.
A002620, {0,1,3}, {1,2,3} , Quarter Squares.
A128422, {0,2,4} , Projective Plane.
A007590, {0,2,6}, {2,4,6}, Arithmetic Mean Triangular Numbers.
A242477, {0,3,9} , Related to quarter squares.
A137932, {0,4,12} , n x n Spiral ?
A004526, {1,1,0}, Non-negative integers repeated.
A001399, {1,1,1} , Partitions at most 3.
A007997, {1,1,2} , Poincare or Molien Series for H^*(S_6, F_2).
A235451, {1,2,0} , Computer Science thing.
A007980, {1,2,4} , Molien Series for H^*(O_3(q); F_2).
A242771, {1,3,5} , Points on Quadralateral.
A000217, {1,3,6} , Triangular Numbers.
A115283, {1,3,9} , Diagonal Sum of Correlation Triangle.
A143978, {1,4,8} , Unit-Squares in a Lattice.
A006578, {1,4,9}, Triangular Numbers + Quarter Squares .
A194275, {1,5,10} , Concentric Pentagonal numbers.
A035608, {1,5,12} , Row Sums of Triangle.
A137719, {1,5,12} , Powers of 2 Hankel Transform.
A198442, {2,3,3} , Constrained Coin Flip.
A001859, {2,3,3} , Triangular Numbers + Quarter Squares ( 2 ) .
A002378, {2,6,12} , oblong numbers, 2 x triangular numbers.

This list probably contains a lot of uninteresting connections, but in
particular I did like to investigate the connection between

A000217, {1,3,6} , Triangular Numbers.
A194275, {1,5,10} , Concentric Pentagonal numbers

The coefficients give a hint to a construction I discovered to pass between
the sequences:

1 is the number of points in a plane left invariant by the actions of
either D3 or D5. 3 or 5 are the number of axes of reflection for a regular
triangle or pentagon. 6 or 10 are the number of elements of group D3 or D5.

In conclusion, these results vastly over specify the vector space and
summarize hundreds or thousands of linear dependences. One connection
between sequences was even sort of interesting. It would be interesting to
expand the search to include multiplication and addition in order to
explore the ring of formal power series rather than a simple vector space.

Cheers,