[seqfan] New sequence.

[Ed Jeffery]

I have a new sequence and would like someone to work out the details (and
proofs) followed by submitting it to OEIS, because I am not able to work on
it. The sequence is described as follows, along with some thoughts on
modifying the definitions of a couple of other existing OEIS sequences.

First, and this is probably a known result, let M_{n,k} be the 2 X 2 matrix

M_{n,k} = [0, 1; k, n].

Prove that the characteristic equation of this matrix is of the form

x^2 - n*x - k = 0,

and with roots a_j (say), j = 1,2, such that

a_1 = (n + sqrt(n^2 + 4*k))/2  and  a_2 = (n - sqrt(n^2 + 4*k))/2.

Second, taking the expressions in the above radical, form the triangle

T(n,k) = n^2 + 4*k, 0 <= k <= n,

which begins as

 0,
 1,  5,
 4,  8, 12,
 9, 13, 17, 21,
16, 20, 24, 28, 32,
25, 29, 33, 37, 41, 45,
36, 40, 44, 48, 52, 56, 60,
...

(This is the sequence to be submitted, since it is not in OEIS.)

Prove or disprove that the row sums of T are given by A007531 [1] (up to an
offset).

Third, because of the definition of T, the definition and offset for
A028884 [2] could be changed to n^2 + 4*(n - 1) with offset 1,2.

Fourth, the outer diagonal {0, 5, 12, ...} of T is A028437 [3] in disguise
so, because of the definition of T, the definition of A028437 could be
changed from n^2 - 4 (offset 2,2) to n^2 + 4*n, or n*(n + 4), with offset
0,2.

Finally (this seems obvious), prove or disprove that the entries of T are
distinct and that A003656 [4] (taken as a set) is a subset of T (taken as a
set).

If someone will submit this sequence, then please let me know the A-number
so I can find it.

Thanks,

LEJ

REFERENCES

[1] OEIS sequence A007531, https://oeis.org/A007531.
[2] OEIS sequence A028884, https://oeis.org/A028884.
[3] OEIS sequence A028437, https://oeis.org/A028347.
[4] OEIS sequence A003656, https://oeis.org/A003656.