possibly wrong comment with Motzkin numbers

Joerg Arndt arndt at jjj.de
Wed Aug 1 08:48:29 CEST 2007 

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The well known euler's sequence n^2-n+41 contains 999 terms < 10^6; 580 of =
them are primes giving a ratio of 58,06%

Studying the matrix :

r=20=20=20=20=20
1          1      3      6      10      15     .      .
2          2      5      9      14         .      .      .
3          4      8     13        .         .      .      .
4          7    12        .        .         .      .      .
5        11       .        .        .         .      .      .

I found that the diagonal r-c=3D399 (r for row, c for colomn) was peculiarl=
y rich in primes.

  r
399    79801      80202      80604
400    80201      80603      81006=20
401    80602      81605      81409

The formula 2*n^2-4*n-197 gives all the numbers of this diagonal for n>=3D2=
01. For smaller values of n we get smaller results, the tens=20

first beeing negative.

The last value < 10^6 is obtained for n=3D708; absolute values of 448 of th=
em are primes giving a ratio of 63,27%.

That is better than the Euler's record, at least in this limit .Moreover th=
e source of this sequence is simpler than Ulam's ones.

So, I am very surprised not to have read anything about it nor in books,nor=
 on the web.

May somebody telle me if this sequence has been already studied and publish=
ed?

I join a complete list of the 708 first terms of the sequence.

As this message is my first contribution, I hope have sent it to the right =
address.

Best regards

Philippe LALLOUET










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<DIV><FONT face=3DArial size=3D2>The well known euler's sequence n^2-n+41 c=
ontains=20
999 terms < 10^6; 580 of them are primes giving a ratio of=20
58,06%</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>Studying the matrix :</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>r     </FONT></DIV>
<DIV><FONT face=3DArial=20
size=3D2>1         =20
1      3     =20
6      10     =20
15     .      .</FONT></D=
IV>
<DIV><FONT face=3DArial=20
size=3D2>2          2&nbs=
p;     5     =20
9     =20
14        =20
.      .      .</FONT></D=
IV>
<DIV><FONT face=3DArial=20
size=3D2>3         =20
4      8     13&nbsp=
;      =20
.         .   &=
nbsp; =20
.      .</FONT></DIV>
<DIV><FONT face=3DArial=20
size=3D2>4         =20
7    12       =20
.       =20
.        =20
.      .      .</FONT></D=
IV>
<DIV><FONT face=3DArial=20
size=3D2>5        11  &nb=
sp;   =20
.       =20
.       =20
.        =20
.      .      .=
</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>I found that the diagonal r-c=3D399 (=
r for row,=20
c for colomn) was peculiarly rich in primes.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>  r</FONT></DIV>
<DIV><FONT face=3DArial size=3D2>399    <FONT color=3D#80000=
0><FONT=20
color=3D#ff0000>79801</FONT> </FONT>    =20
80202      80604</FONT></DIV>
<DIV><FONT face=3DArial size=3D2>400   =20
80201      <FONT=20
color=3D#ff0000>80603  </FONT>    81006 </FONT></D=
IV>
<DIV><FONT face=3DArial size=3D2>401   =20
80602      81605    &nbsp=
;=20
<FONT color=3D#ff0000>81409</FONT></FONT></DIV>
<DIV><FONT face=3DArial color=3D#ff0000 size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>The formula 2*n^2-4*n-197 gives all the nu=
mbers of=20
this diagonal for n>=3D201. For smaller values of n we get smaller resul=
ts, the=20
tens </FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>first beeing negative.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>The last value < 10^6 is obtained for n=
=3D708;=20
absolute values of 448 of them are primes giving a ratio of 63,27%.</FONT><=
/DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>That is better than </FONT><FONT face=3DAr=
ial=20
size=3D2>the Euler's record, at least in this limit .Moreover the source of=
 this=20
sequence is simpler than Ulam's ones.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>So, I am very surprised not to have read a=
nything=20
about it nor in books,nor on the web.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>May somebody telle me if this sequence has=
 been=20
already studied and published?</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>I join a complete list of the 708 first te=
rms of=20
the sequence.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>As this message is my first contribution, =
I hope=20
have sent it to the right address.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>Best regards</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>Philippe LALLOUET</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
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