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%I #61 Nov 18 2023 11:46:31
%S 0,1,2,1,2,1,2,0,0,0,0,1,2,1,2,1,0,0,0,0,0,0,0,0,0,1,2,1,2,1,2,1,2,1,
%T 2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,
%U 2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2
%N Let d = A307720(n) - A307720(n-1); a(n) = 0 if d = 0; a(n) = 1 if d > 0; a(n) = 2 if d < 0.
%C If one looks at the graph of A307720 (that entry has a number of versions of this graph besides the one that appears when the "graph" button is clicked), one sees that initially A307720(n) is usually greater than A307720(n-1) if n is odd.
%C Think of A307720 as the a piano score in which normally the right hand (n = 2*i+1) is to the right of the left hand (n = 2*i).
%C However, as can be seen in _William Cheswick_'s colored plots, sometimes the right and left hands swap. In these plots, the right-hand points (n odd) are blue and the left-points (n even) are red.
%C A run of terms 121212... in the present sequence is a normal sequence of notes left, right, left, right, ..., where blue is on top.
%C A run 212121... indicates that the hands have been swapped (red is on top).
%C A run 000000... indicates that both hands are playing the same note (red and blue are at the same level.
%C The purpose of the present sequence and related sequences is to study when the hands swap. At present there is no explanation for when this happens.
%C The sequence of pictures suggests that there will be infinitely many occasions when the hands swap. The upper color in the picture will alternate infinitely often between red and blue, with longer and longer runs before the upper color changes.
%H William Cheswick, <a href="/A348248/a348248_2.png">Colored plot of first 200 terms of A307720</a> (See Comments for explanation of colors in these pictures)
%H William Cheswick, <a href="/A348248/a348248_3.png">Colored plot of first 1000 terms of A307720</a>
%H William Cheswick, <a href="/A348248/a348248_4.png">Colored plot of first 10000 terms of A307720</a>
%H William Cheswick, <a href="/A348248/a348248.png">Colored plot of first 10^5 terms of A307720</a>
%H William Cheswick, <a href="/A348248/a348248_1.png">Colored plot of first 10^6 terms of A307720</a>
%H Rémy Sigrist, <a href="/A307720/a307720.png">Scatterplot of the first 10000000 terms of A307720</a>
%H Chai Wah Wu, <a href="/A348446/a348446_2.png">Scatterplot of the first 100 million terms of A348446</a> [Shows how the lead changes between the left and right hands. The color scheme here is different from that in _William Cheswick_'s pictures.]
%Y Cf. A307720, A307730, A348446, A348459.
%K nonn
%O 2,3
%A _N. J. A. Sloane_, Oct 20 2021
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