[seqfan] Re: An idea of sequence of sequences concerning with triangle

Vladimir Shevelev shevelev at bgu.ac.il
Sun Nov 9 13:58:53 CET 2014


 In addition, note that, if to use sequence {a(n)}, we can consider it arbitrary nonnegative.
Indeed, if a_k=1 ("every first entry is replaced by opposite"), then, on the
k-th step, all 1's (0's) are changed by 0's (1's). Besides, we can agree that,
if a_k=0, then all entries are changed by 0's.
 It is interesting to create a calculator such that, when the user inputs several 
nonnegative numbers (e.g., 3,2,4,3,5,1), on the screen there apears
the last sequence and, maybe, the last sequence of row sums (e.g., 20-30 terms). Besides, one can create a picture of several first rows of triangle in the last its 
state, where 0's and 1's are represented by white and red small circles respectively.

Best regards,
Vladimir

________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 08 November 2014 17:58
To: seqfan at list.seqfan.eu
Subject: [seqfan] An idea of sequence of sequences concerning with triangle

Dear seqfans,

Let us start with triangle of 1's

1
11
111
1111
....

Reading it in binary and converting to decimal, we obtain
Mersenne numbers (A000225): 1,3,7,15,31,63,127,...
Numbering 1's in triangle consecutively, let us change
every second 1's by 0. We get triangle

1
01
010
1010
....
Again converting rows to decimal, we have sequence
1,1,2,10,21,21,42,...
Further, we change every third entry in the second triangle
by opposite (i.e., 1->0, 0->1):
1
00
011
1000
....
and get sequence  1,0,3,8,28,28,56,...
Changing here every fourth entry by opposite , we get
sequence 1,0,7,12,20,62,41,..., etc.
Instead of using the sequence "second,third,fourth,...",
we can use "a_2-th, a_3-th,..." for any sequence {a_n(>1)}.

Best regards,
Vladimir

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