duplicates

[Rob Pratt]

Neil,

For all three of these, the correct year is 2005 (not 2002).

Rob  

-----Original Message-----
From: N. J. A. Sloane [mailto:njas at research.att.com] 
Sent: Monday, June 13, 2005 9:24 PM
To: seqfan; Christian G.Bower
Cc: bowerc at usa.net; njas at research.att.com
Subject: Re: duplicates

Concerning Christian's message, I cannot resist mentioning three sequences that I added to the OEIS yesterday - a strange
coincidence!   Look:

%S A107755 2,8,12,26,30,36,38,80,84,90,92 %N A107755 Numbers n such that Sum_{k=1..n} Catalan(k) == 0 mod 3.
%D A107755 Y. More, Problem 11165, Amer. Math. Monthly, 112 (2002), 568.

%S A107756 1,4,5,6,7,10,13,14,15,16,17,18,19,20,21,22,23,24,25,28,31,32,33,34,37,
%T A107756 40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,
%U A107756 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,82,85,86,87,88,91,94,95
%N A107756 Numbers n such that Sum_{k=1..n} Catalan(k) == 1 mod 3.
%D A107756 Y. More, Problem 11165, Amer. Math. Monthly, 112 (2002), 568.
%F A107756 Equivalently, numbers n such that base 3 expansion of n+1 contains a 2.
!!!!!!

%S A107757 3,9,11,27,29,35,39,81,83,89,93 %N A107757 Numbers n such that Sum_{k=1..n} Catalan(k) == 2 mod 3.
%D A107757 Y. More, Problem 11165, Amer. Math. Monthly, 112 (2002), 568.

I don't know if this is anything more than a coincidence.   NJAS

>From seqfan-owner at ext.jussieu.fr  Mon Jun 13 00:12:01 2005
>Date: Sun, 12 Jun 2005 21:09:41 -0700
>From: "Christian G.Bower" <bowerc at usa.net>
>To: seqfan <seqfan at ext.jussieu.fr>
>Subject: duplicates

>I'm looking at one that someone on the list might help with
>
>1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,1
>A039966 An example of a d-perfect sequence. 
>A104405 Number of partitions of n into distinct powers of 3. 
>??
>
>ID Number: A104405
>URL:       http://www.research.att.com/projects/OEIS?Anum=A104405
>Sequence:  1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,
>           1,0,0,0,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
>           0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0,0,1,
>           1,0,1,1,0,0,0,0,0,0,0,0,0,0,0
>Name:      Number of partitions of n into distinct powers of 3.
>
>can be described as characteristic function of numbers with no "2"   !!!!!
>in their ternary representation
>
>ID Number: A039966
>URL:       http://www.research.att.com/projects/OEIS?Anum=A039966
>Sequence:  1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,
>           1
>Name:      An example of a d-perfect sequence.
>
>can be described as A005043(n-1) mod 3
>
>ID Number: A005043 (Formerly M2587)
>URL:       http://www.research.att.com/projects/OEIS?Anum=A005043
>Sequence:  1,0,1,1,3,6,15,36,91,232,603,1585,4213,11298,30537,83097,
>           227475,625992,1730787,4805595,13393689,37458330,105089229,
>           295673994,834086421,2358641376,6684761125,18985057351,
>           54022715451,154000562758
>Name:      Motzkin sums: a(n) = (n-1)*(2*a(n-1)+3*a(n-2))/(n+1). Also called
>              Riordan numbers or ring numbers.
>
>I've verified the 2 sequences match for the first 512 terms.
>
>Christian
>
>
>