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Positions inventory sequence: for stage k >= 2 we record where all the numbers from the two previous stages have appeared, starting with a(0) = 0, a(1) = 0.
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%I #35 Nov 22 2022 22:18:09

%S 0,0,0,1,1,2,3,2,3,4,5,6,4,5,7,6,8,9,10,11,7,8,9,12,10,13,11,15,14,16,

%T 17,18,19,12,13,15,14,20,16,21,17,22,18,24,19,26,23,25,28,27,29,30,31,

%U 32,20,21,22,24,26,23,33,25,34,28,36,27,35,29,38,30,40,31,42

%N Positions inventory sequence: for stage k >= 2 we record where all the numbers from the two previous stages have appeared, starting with a(0) = 0, a(1) = 0.

%C When the sequence is displayed as a triangle, the row corresponding to stage k >= 2 is a permutation of the numbers from Fibonacci(k) - 1 to Fibonacci(k+2) - 2.

%e As an irregular triangle, the sequence begins:

%e 0;

%e 0;

%e 0, 1;

%e 1, 2, 3;

%e 2, 3, 4, 5, 6;

%e 4, 5, 7, 6, 8, 9, 10, 11;

%e 7, 8, 9, 12, 10, 13, 11, 15, 14, 16, 17, 18, 19;

%e ...

%e At stage 5 we look at two previous stages 3 and 4 and see the

%e positions of 1's: 4;

%e positions of 2's: 5, 7;

%e positions of 3's: 6, 8;

%e positions of 4's: 9;

%e positions of 5's: 10;

%e positions of 6's: 11;

%e thus stage 5 is 4, 5, 7, 6, 8, 9, 10, 11.

%Y Cf. A000045, A000071, A001911, A356784, A357317, A357443.

%K nonn,tabf

%O 0,6

%A _Ctibor O. Zizka_, Nov 21 2022