creigh at o2online.de creigh at o2online.de
Mon Sep 6 17:28:02 CEST 2004

```Just found an additional property (VIII) which perplexes
me. I also fixed a mistake at the end.

*********************

Apparently, none of the four sequences

jes seq:  1(0), -4(1), 15(2), -56(3), 209(4), -780(5), 2911(6), -10864(7),
40545(8), -151316(9), 564719(10)

les seq:  -2(0), 1(1), -6(2), 16(3), -62(4), 225(5), -842(6), 3136(7), -11706
(8), 43681(9), -163022(10)

tes seq:  -1(0), 4(1), -13(2), 49(3), -181(4), 676(5), -2521(6), 9409(7), -35113
(8), 131044(9), -489061(10)

ves seq: -2(0), 1(1), -4(2), 9(3), -34(4), 121(5), -452(6), 1681(7), -6274(8),
23409(9), -87364(10)

are listed at OEIS. However, notice the symmetries (for all n):

I:   jes(n) + les(n) + tes(n) = ves(n)

II:  les(2n+1); tes(2n+1); ves(2n+1);
ves(2n+1) - jes(2n+1) - 1 =  les(2n+1) +  tes(2n+1) - 1;
3*les(2n+1) + 1 = 3*jes(n)^2 + 1 (see IV and VII)  are perfect squares

III:  les(2n+1) divides ves(2n+1) - jes(2n+1) - 1 =  les(2n+1) +  tes(2n+1) -
1

IV: (jes(n))^2 = les(2n+1)

V: tes(2n) =  A001570(n), sqrt(  tes(2n+1)  ) = A001075(n)  [both are "nice"
sequences]

VI: ves(2n), ves(2n)/2 do not exist in OEIS, however: sqrt(  ves(2n+1)  )
= A001835(n) [ "nice" ]

VII: sqrt(  les(2n+1)  ) = A001353(n) [ Comments:  3*a(n)^2 + 1 is a perfect square.
]

VIII: les(n) + tes(n) = ves(2+n) + jes(n)

It follows immediately from properties I + II  that...
1^2 (les(1) + 2^2 (tes(1))  = 2^2 (ves(1) - jes(1) - 1)
4^2 (les(3))+ 7^2 (tes(3))  = 8^2 (ves(3) - jes(3) - 1)
15^2 (les(5)) + 26^2 (tes(5))  = 30^2 (ves(5) - jes(5) - 1)

Sincerely,
Creighton

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