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Infinite triangle T(n, k), n, k >= 0, read and filled by rows the greedy way with distinct integers such that for any n, k >= 0, T(n, k) + T(n+1, k) + T(n+1, k+1) = 0; each term is minimal in absolute value and in case of a tie, preference is given to the positive value.
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%I #14 Mar 14 2023 03:44:10

%S 0,1,-1,2,-3,4,3,-5,8,-12,5,-8,13,-21,33,6,-11,19,-32,53,-86,-2,-4,15,

%T -34,66,-119,205,9,-7,11,-26,60,-126,245,-450,10,-19,26,-37,63,-123,

%U 249,-494,944,7,-17,36,-62,99,-162,285,-534,1028,-1972

%N Infinite triangle T(n, k), n, k >= 0, read and filled by rows the greedy way with distinct integers such that for any n, k >= 0, T(n, k) + T(n+1, k) + T(n+1, k+1) = 0; each term is minimal in absolute value and in case of a tie, preference is given to the positive value.

%C Will every integer appear in the triangle?

%H Rémy Sigrist, <a href="/A361442/b361442.txt">Table of n, a(n) for n = 0..10010</a>

%H Rémy Sigrist, <a href="/A361442/a361442.png">Colored representation of the first 500 rows</a> (the color is function of the sign of T(n, k))

%H Rémy Sigrist, <a href="/A361442/a361442.gp.txt">PARI program</a>

%F T(n, 0) = A361443(n).

%F T(n, k) = (-1)^k * Sum_{i = 0..k} binomial(k, i) * T(n-i, 0).

%e Triangle begins:

%e 0

%e 1 -1

%e 2 -3 4

%e 3 -5 8 -12

%e 5 -8 13 -21 33

%e 6 -11 19 -32 53 -86

%e -2 -4 15 -34 66 -119 205

%e 9 -7 11 -26 60 -126 245 -450

%e 10 -19 26 -37 63 -123 249 -494 944

%e 7 -17 36 -62 99 -162 285 -534 1028 -1972

%o (PARI) See Links section.

%Y Cf. A152920, A361443.

%K sign,tabl

%O 0,4

%A _Rémy Sigrist_, Mar 12 2023