Do any integers occur in both sequences?

[Peter Pein]

Well, I should not post as early in the morning (at least not before
I've got the third cup of coffee). I did not add a(n-1)....

Sorry for any inconvenience, trouble and the like

Peter

P.S.:
A131938-A131937 is (seems to be after coffee ;) ):

{1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6,
  7, 8, 8, 8, 8, 8, 8, 9, 10, 11, 11, 11, 11,
  11, 11, 12, 13, 14, 15, 15, 15, 15, 15, 15,
  15, 16, 17, 18, 19, 19, 19, 19, 19, 19, 19,
  19, 20, 21, 22, 23, 23, 23, 23, 23, 23, 23,
  23, 23, 24, 25, 26, 27, 27, 27, 27, 27, 27,
  27, 27, 27, 27, 28, 29, 30, 31, 32, 32, 32,
  32, 32, 32, 32, 32, 32, 32, 33, 34, 35, 36,
  37, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38,
  38, 39, 40, 41, 42, 43, 44, 44, 44, 44, 44,
  44, 44, 44, 44, 44, 44, 44, 45, 46, 47, 48,
  49, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50,
  50, 50, 50, 51, 52, 53, 54, 55, 56, 56, 56,
  56, 56}

Peter Pein schrieb:
> Leroy Quet schrieb:
>> I have just submitted these two interdependent sequences (So don't look 
>> for them in the database yet):
>>
>>> %I A131937
>>> %S A131937 1,4,8,14,21,29,38,49,61
>>> %N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which 
>>> does not occur in sequence A131938).
>>> %e A131937 A131938: 2,5,10,16,23,32,42,53,...
>>> Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
>>> So A131937(8) = A131937(7) + 11 = 49.
>>> %Y A131937 A131938
>>> %O A131937 1
>>> %K A131937 ,more,nonn,
>>> %I A131938
>>> %S A131938 2,5,10,16,23,32,42,53,65,78,93,109
>>> %N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which 
>>> does not occur in sequence A131937).
>>> %e A131938 A131937: 1,4,8,14,21,29,...
>>> Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
>>> So A131938(8) = A131938(7) + 11 = 53.
>>> %Y A131938 A131937
>>> %O A131938 1
>>> %K A131938 ,more,nonn,
>>
>> I have not thought about this too hard; so for all I know, the proof is 
>> quite easy.
>>
>> Do any positive integers occur in both A131937 and A131938?
>>
>> Thanks,
>> Leroy Quet
>>
> Hi again,
> 
> even if you meant "n-th pos. int. which does yet not occur in either l1
> or l2" (or does one say "..does yet neither occur in l1 nor in l2"?) I
> get another result
> 
> list1={1,4};list2={2,5};n=3;
> Do[
>     Print[{list1,list2}];
>     Print["n= ",n," ",Complement[Range[25],list2]];
>     AppendTo[list1,Part[Complement[Range[3n ],Union[list1,list2]],n]];
>     Print["n= ",n," ",Complement[Range[25],list1]];
>     AppendTo[list2,Part[Complement[Range[3n],Union[list1,list2]],n]];
>     n++;
>     ,{5}];
> list1
> list2
> Intersection[%%,%]
> 
....



I wrote:
>I have just submitted these two interdependent sequences (So don't look 
>for them in the database yet):
>
>>%I A131937
>>%S A131937 1,4,8,14,21,29,38,49,61
>>%N A131937 a(1)=1; a(2)=4. a(n) = a(n-1) + (nth positive integer which 
>>does not occur in sequence A131938).
>>%e A131937 A131938: 2,5,10,16,23,32,42,53,...
>>Positive integers not in A131938: 1,3,4,6,7,8,9,11,...
>>So A131937(8) = A131937(7) + 11 = 49.
>>%Y A131937 A131938
>>%O A131937 1
>>%K A131937 ,more,nonn,
>
>>%I A131938
>>%S A131938 2,5,10,16,23,32,42,53,65,78,93,109
>>%N A131938 a(1)=2; a(2)=5. a(n) = a(n-1) + (nth positive integer which 
>>does not occur in sequence A131937).
>>%e A131938 A131937: 1,4,8,14,21,29,...
>>Positive integers not in A131937: 2,3,5,6,7,9,10,11,...
>>So A131938(8) = A131938(7) + 11 = 53.
>>%Y A131938 A131937
>>%O A131938 1
>>%K A131938 ,more,nonn,
>
>I have not thought about this too hard; so for all I know, the proof is 
>quite easy.
>
>Do any positive integers occur in both A131937 and A131938?
>
>Thanks,
>Leroy Quet

I have made no progress towards determining if any particular positive 
integers occur within both sequences. (In other words: Does  A131937(k) = 
A131938(j) for any j and k {j and k are >= 1}, where j need not equal k?)

I conjecture that no particular positive integer occurs in both sequences.


Here is a smaller result related to these sequences, which I doubt will 
help (dis)prove the main conjecture:

Let a(n) =  A131937(n), b(n) = A131938(n), a(0) = b(0) = 0.

Let n = any positive integer.

Then n occurs (a(n) - a(n-1) - 1) times in sequence {b(n) - b(n-1) - n + 
1}.

And n occurs (b(n) - b(n-1) - 1) times in sequence {a(n) - a(n-1) - n + 
1}.

Not too earth-shattering -- but while we're on the subject...


Also, I wonder if anyone can come up with a closed form for {A131937(n)} 
and {A131938(n)}.
They seem like they might be related to Beautty sequences somehow.

Thanks,
Leroy Quet



A comment with seq A001006
http://www.research.att.com/~njas/sequences/A001006

This seems to be wrong, I get the counts

Note the starts math, Motzkins start as


Someone please verify!