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Triangle read by rows: n-th diagonal (from the right) is the sequence of (signed) differences between pairs of consecutive terms in the (n-1)th diagonal. The rightmost 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.
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%I #12 May 31 2017 07:57:52

%S 1,2,3,4,6,9,-5,-1,5,14,13,8,7,12,26,-30,-17,-9,-2,10,36,75,45,28,19,

%T 17,27,63,-200,-125,-80,-52,-33,-16,11,74,524,324,199,119,67,34,18,29,

%U 103,-1299,-775,-451,-252,-133,-66,-32,-14,15,118

%N Triangle read by rows: n-th diagonal (from the right) is the sequence of (signed) differences between pairs of consecutive terms in the (n-1)th diagonal. The rightmost 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 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 The triangle begins:

%e 1,

%e 2,3,

%e 4,6,9,

%e -5,-1,5,14,

%e 13,8,7,12,26,

%e -30,-17,-9,-2,10,36.

%e Example:

%e Considering the rightmost value of the 4th row: Writing a 10 here instead, the first 4 rows of the triangle become:

%e 1

%e 2,3

%e 4,6,9

%e -9,-5,1,10

%e But 1 already occurs earlier in the triangle. So 10 is not the rightmost element of row 4.

%e Checking 11,12,13,14; 14 is the smallest value that can be the rightmost element of row 4 and not have any elements of row 4 occur earlier in the triangle.

%Y Cf. A035312, A136562, A136563.

%K sign,tabl

%O 1,2

%A _Leroy Quet_, Jan 06 2008

%E Rows 7-10 from _Andrey Zabolotskiy_, May 29 2017