# [seqfan] Re: suggestion: new keywords "polynomial", "rational", "conj.polynomial", "conj.rational"

andre maute andre.maute at gmx.de
Thu Feb 26 13:32:35 CET 2009

```>I would like to suggest the following 4 additional keywords
>
>"polynomial",
>there is a polynomial function p(n) producing the sequence
>
>"rational",
>there is a rational function p(n)/q(n) producing the sequence
>
>"conj-polynomial",
>there could exist a polynomial function p(n) producing the sequence
>
>"conj-rational"
>there could exist a rational function p(n)/q(n) producing the sequence
>
>The conj-variants, meaning conjecture, could be applied if e.g.
>a computer algebra script could collect enough evidence.

I had the following problem, let p >= 1 be a natural number, then compute

the number of 4-tuples with
{ (a1,a2,b1,b2) natural numbers with
0 <= a1+a2 <= p, 0 <= b1+b2 <= p,
abs((a1+a2)-(b1+b2)) <= 3, abs(a2-b2) <= 2 }

i wrote a script and got

9, 36, 92, 181, 305, 464, 658, 887, 1151, 1450, 1784,
2153, 2557, 2996, 3470, 3979, 4523, 5102, 5716, 6365,
7049, 7768, 8522, 9311, 10135, 10994, 11888, 12817,
13781, 14780, 15814, 16883, 17987, 19126, 20300, 21509

now combining 3 consecutive numbers and interpolating
one gets the following polynomials

(9,36,92) --> 29*x^2/2-33*x/2+11,
(36,92,181) --> 33*x^2/2-53*x/2+23,

for all other cases

35*x^2/2-67*x/2+35,

perhaps i should have been more concrete and add to my suggestions
with p(n), q(n) polynomials with small degree say deg(p),deg(q) <= 10

Regards
Andre

```