# puzzle

Meeussen Wouter (bkarnd) wouter.meeussen at vandemoortele.com
Wed Mar 3 13:40:00 CET 2004

```the integer digits base 3, of the first are

{{0}, {1}, {2}, {1, 1}, {1, 2}, {2, 2}, {1, 1, 1}, {1, 1, 2}, {1, 2, 2},

{2, 2, 2}, {1, 1, 1, 1}, {1, 1, 1, 2}, {1, 1, 2, 2}, {1, 2, 2, 2}, {2,
2, 2, 2}, ...

while twice the second gives:
{{2}, {1, 1}, {2, 0}, {1, 0, 1}, {1, 1, 0}, {2, 0, 0}, {1, 0, 0, 1}, {1,
0, 1, 0}, {1, 1, 0, 0},
{2, 0, 0, 0}, {1, 0, 0, 0, 1}, {1, 0, 0, 1, 0}, {1, 0, 1, 0, 0}, {1, 1,
0, 0, 0}, {2, 0, 0, 0, 0}

so the first is 1 everywhere but with a step-up to 2 along the way.
Now read it left to right :
(-1 + 3^(p))/(3 - 1)   +   (-1 + 3^(q))/(3 - 1)

The second is (3^u + 3^v)/2 - 1,
or zero everywhere with   'spikes' at position u and v.

FullSimplify[(3^(u) + 3^(v))/2 -
1 == (-1 + 3^(p))/(3 - 1) + (-1 + 3^(q))/(3 - 1)]

gives
3^p + 3^q == 3^u + 3^v

so p=u and q=v is a solution.

What have I dropped along the way?

Wouter Meeussen
Senior Scientist
R&D dept.
Lipids & Dough div.
Vandemoortele NV
+32 (0) 51 33 21 24 tel
+32 (0) 51 33 21 75 fax

-----Original Message-----
From: Ralf Stephan [mailto:ralf at ark.in-berlin.de]
Sent: woensdag 3 maart 2004 12:13
To: seqfan at ext.jussieu.fr
Subject: puzzle

Can you prove or disprove that A073216 = A023745 + 1?

%S A023745
0,1,2,4,5,8,13,14,17,26,40,41,44,53,80,121,122,125,134,161,242,364,
%N A023745 Plaindromes: digits in base 3 are in ascending order.

%S A073216
1,2,3,5,6,9,14,15,18,27,41,42,45,54,81,122,123,126,135,162,243,365,
%N A073216 The terms of A055235 (sums of two powers of 3) divided by 2.

I must have myopia but I will write the database entry for you if
you show the beef.

ralf

===============================
This email is confidential and intended solely for the use of the individual to whom it is addressed.
If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited.
You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

```