[seqfan] Re: Formulas needed for new sequence.

[Paolo Lava]

Dear Ed,

A191748={1,2,5,6,10,14,...},

a(n+1)=a(n)+(1-b)*c+b, where b is one of the formula in A010054 and c one of
the formula in A002024. Of course you need to correct the offset.

For example, taking the formula of Carl R. White in A010054 and the formula
of Neil Sloane in the comment section of A002024 you have:

a(n+1)=a(n)+(1-floor((1-cos(Pi*sqrt(8*(n-1)+1)))/2))*floor((1+sqrt(1+8*(n-2)))/2+1)+floor((1-cos(Pi*sqrt(8*(n-1)+1)))/2)

This is not a closed form but just a first attempt to find a formula:
probably there is something less contrived than this...

Paolo P. Lava

2011/6/29 Ed Jeffery <ed.jeffery at yahoo.com>

> Fellow sequence fanatics,
>
> I recently submitted the following seemingly trivial sequence to OEIS:
>
> A191747={1,1,0,0,1,1,0,0,0,1,0,0,0,1,...} (offset=1).
>
> This sequence is just a concatenation of row entries of the N X N identity
> matrices and hasn't been reviewed yet.
>
> However, of interest is the sequence of all m in {1,2,3,...} for which
> A191747(m)=1. This definition gives the sequence
>
> A191748={1,2,5,6,10,14,...},
>
> which can be read by anti-diagonals from the table
>
> 1,  5,  14,  30,  55,  ..
> 2,  10, 25,  49,  84,  ..
> 6,  20, 43,  77,  124, ..
> 15, 37, 70,  116, 177, ..
> 31, 63, 108, 168, 245, ..
> etc..
>
> I haven't submitted this sequence yet, since I am in the midst of some
> adversity and need help with a formula for the general term: I haven't been
> able to concentrate long enough to get this done myself, so if some of you
> would like to help, it would be greatly appreciated.
>
> Some analysis:
>
> The first row of the table is the core sequence A000330, omitting the
> initial 0, and the first column is A056520. The j-th row can be found from
> the generating functions
>
> (1)  (-j+(2*j+1)*x-(j-1)*x^2)/(1-x)^4, j in {0,1,2,...},
>
> by taking from the (j+1)-th term on, but this contradicts the definition of
> A000330 (corresponding to j=0) which has the initial term set to zero,
> although the generating functions are in agreement: the reason for the
> initial zero evidently is the alternative definition
>
> A056520(n)=A000300(n)+1.
>
> Here, j=1 in (1) gives essentially A058373, ignoring the two initial zeros.
> (A058373, by the way, was defined with offset=-2 which is bewildering to me,
> and perhaps it should be changed to offset=0 after dropping the two zeros.)
> Putting j=2 gives -A058372 which leads me to believe that the signs should
> be reversed, that is, A058372 should be negated and rewritten in the
> database. As far as I can determine, none of the other rows or columns of
> the table appear in the OEIS database.
>
> Continuing with analysis of the above table, similarly, the k-th column can
> be found from the generating functions
>
> (2*k+1-(5*k+2)*x+4*(k+1)*x^2-(k+1)*x^3)/(1-x)^4, k in {0,1,2,...},
>
> by taking the k-th term on.
>
> Likewise, for j and k as above, for the j-th row R_j, we have the
> closed-form expression
>
> R_j(n)=(n+1)*(2*n^2+n-6*j)/6, n=j+1,j+2,j+3,...;
>
> and for the k-th column C_k,
>
> C_k(n)=(n+2)*(2*n^2-n+6*k+3), n=j,j+1,j+2,....
>
> In conclusion, I see no way of defining a generating function for this
> sequence, even though the rows and columns have generating functions with a
> common denominator, namely (1-x)^4. Maybe you all would like to try to find
> this function, if it exists. Furthermore, I suspect that a closed-form
> expression for a general term of the sequence exists, but I haven't been
> able to work it out. At present, my alleged mind can't handle complications
> with multiple indices (j and k) and varying offsets. So, I'll leave it to
> those with more experience and hope to get a unified expression soon, at
> which point I'll go ahead and submit the sequence.
>
> Thanks in advance to everyone,
>
> Ed Jeffery
>
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