A Non-Duplicating Difference Triangle

[Leroy Quet]

I just submitted the following sequences:

%I A136561
%S A136561
1,2,3,4,6,9,-6,-2,4,13,16,10,8,12,25,-41,-25,-15,-7,5,30
%N A136561 Triangle: nth diagonal (from the
right) is the sequence of (signed) differences
between pairs of consecutive terms in the (n-1)th
diagonal. The right-most diagonal (A136562) if
defined: A136562(1)=1; A136562(n) is the smallest
integer > A136562(n-1) such that any (signed)
integer occurs at most once in the triangle
A136561.
%C A136561 Requiring that the ABSOLUTE values of
the differences in the difference triangle only
occur at most once each leads to the Zorach
additive triangle. (See A035312.)
%e A136561 The triangle begins:
1
2,3,
4,6,9,
-6,-2,4,13
16,10,8,12,25
-41,-25,-15,-7,5,30.
Example:
Considering the right-most value of the 4th row.
Writing a 10 here instead, the first 4 rows of
the triangle becomes:
1
2,3
4,6,9
-9,-5,1,10
But 1 already occurs earlier in the triangle. So
10 is not the right-most element of row 4.
Checking 11,12,13; 13 is the smallest value that
can be the right-most element of row 4 and not
have any elements of row 4 occur earlier in the
triangle.
%Y A136561 A035312,A136562,A136563
%O A136561 1
%K A136561 ,more,sign,tabl,

%I A136562
%S A136562 1,3,9,13,25,30
%N A136562 Consider the triangle A136561: the nth
diagonal (from the right) is the sequence of
(signed) differences between pairs of consecutive
terms in the (n-1)th diagonal. The right-most
diagonal (A136562) is defined: A136562(1)=1;
A136562(n) is the smallest integer >
A136562(n-1) such that any (signed) integer
occurs at most once in the triangle A136561.
%C A136562 Requiring that the ABSOLUTE values of
the differences in the difference triangle only
occur at most once each leads to the Zorach
additive triangle. (See A035312.) The rightmost
diagonal of the Zorach additive triangle is
A035313.
%e A136562 The triangle A136561 begins:
1
2,3,
4,6,9,
-6,-2,4,13
16,10,8,12,25
-41,-25,-15,-7,5,30.
Example:
Considering the right-most value of the 4th row:
Writing a 10 here instead, the first 4 rows of
the triangle become:
1
2,3
4,6,9
-9,-5,1,10
But 1 already occurs earlier in the triangle. So
10 is not the right-most element of row 4.
Checking 11,12,13; 13 is the smallest value that
can be the right-most element of row 4 and not
have any elements of row 4 occur earlier in the
triangle. So A136562(4) = 13.
%Y A136562 A035313,A136561,A136563
%O A136562 1
%K A136562 ,more,nonn,

%I A136563
%S A136563 1,2,4,-6,16,-41
%N A136563 Leftmost column of triangle A136561.
%C A136563 Requiring that the ABSOLUTE values of
the differences in the difference triangle only
occur at most once each leads to the Zorach
additive triangle. (See A035312.) The leftmost
column of the Zorach additive triangle is
A035311.
%Y A136563 A035311,A136561,A136562
%O A136563 1
%K A136563 ,more,sign,

First, I find it hard to believe these sequences
are not yet in the EIS. Did I make a stupid
error? (I didn't even use a calculator.)

Second, there is a comment at A035312 that the
terms of A035312 are conjectured to form a
permutation of the positive integers.
I will ask then, are the terms of A136561 a
permutation of all non-zero integers?

Finally, am I missing something, or does the
comment at A035313 about the difference triangle
miss the fact that we are talking about ABSOLUTE
differences in regards to the Zorach additive
triangle?

Thanks,
Leroy Quet



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