A348248 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.

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, 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, 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

Offset

2,3

Comments

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.

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).

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.

A run of terms 121212... in the present sequence is a normal sequence of notes left, right, left, right, ..., where blue is on top.

A run 212121... indicates that the hands have been swapped (red is on top).

A run 000000... indicates that both hands are playing the same note (red and blue are at the same level.

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.

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.

Links

Table of n, a(n) for n=2..110.

William Cheswick, Colored plot of first 200 terms of A307720 (See Comments for explanation of colors in these pictures)

William Cheswick, Colored plot of first 1000 terms of A307720

William Cheswick, Colored plot of first 10000 terms of A307720

William Cheswick, Colored plot of first 10^5 terms of A307720

William Cheswick, Colored plot of first 10^6 terms of A307720

Rémy Sigrist, Scatterplot of the first 10000000 terms of A307720

Chai Wah Wu, Scatterplot of the first 100 million terms of A348446 [Shows how the lead changes between the left and right hands. The color scheme here is different from that in William Cheswick's pictures.]

Crossrefs

Cf. A307720, A307730, A348446, A348459.

Sequence in context: A323284 A010250 A268041 * A060024 A143668 A029445

Adjacent sequences: A348245 A348246 A348247 * A348249 A348250 A348251

Keyword

nonn

Author

N. J. A. Sloane, Oct 20 2021

Status

approved