A362913 Array of numbers read by upward antidiagonals: leading row lists phi(i), i >= 1 (cf. A000010); the following rows give absolute values of differences of previous row.

1, 0, 1, 1, 1, 2, 0, 1, 0, 2, 1, 1, 2, 2, 4, 0, 1, 2, 0, 2, 2, 1, 1, 0, 2, 2, 4, 6, 0, 1, 2, 2, 0, 2, 2, 4, 1, 1, 2, 0, 2, 2, 0, 2, 6, 0, 1, 2, 0, 0, 2, 0, 0, 2, 4, 1, 1, 0, 2, 2, 2, 4, 4, 4, 6, 10, 0, 1, 2, 2, 0, 2, 4, 0, 4, 0, 6, 4, 1, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 8, 12, 0, 1, 2, 2

Offset

1,6

Comments

The leading entries in the rows (the Gilbreath transform of A000010, cf. A362451) appear to form the period-2 sequence 1,0,1,0,1,0,... Is there a simple proof? This would follow if there was a proof that the Gilbreath transform of |A057000| is the all-1's sequence.

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)

N. J. A. Sloane, Maple code for Gilbreath transform and related arrays

N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)

Example

The array begins:
  1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, ...
  0, 1, 0, 2, 2, 4, 2, 2, 2, 6, 6, 8, 6, 2, 0, ...
  1, 1, 2, 0, 2, 2, 0, 0, 4, 0, 2, 2, 4, 2, ...
  0, 1, 2, 2, 0, 2, 0, 4, 4, 2, 0, 2, 2, ...
  1, 1, 0, 2, 2, 2, 4, 0, 2, 2, 2, 0, ...
  0, 1, 2, 0, 0, 2, 4, 2, 0, 0, 2, ...
  1, 1, 2, 0, 2, 2, 2, 2, 0, 2, ...
  0, 1, 2, 2, 0, 0, 0, 2, 2, ...
  1, 1, 0, 2, 0, 0, 2, 0, ...
  ...
The first few antidiagonals are:
  1
  0, 1
  1, 1, 2
  0, 1, 0, 2
  1, 1, 2, 2, 4
  0, 1, 2, 0, 2, 2
  1, 1, 0, 2, 2, 4, 6
  0, 1, 2, 2, 0, 2, 2, 4
  1, 1, 2, 0, 2, 2, 0, 2, 6
  0, 1, 2, 0, 0, 2, 0, 0, 2, 4
  ...

Mathematica

A362913[dmax_]:=With[{d=Reverse[NestList[Abs[Differences[#]]&, EulerPhi[Range[dmax]], dmax-1]]}, Array[Diagonal[d, #]&, dmax, 1-dmax]]; A362913[20] (* Generates 20 antidiagonals *) (* Paolo Xausa, May 10 2023 *)

Crossrefs

Cf. A000010 (top row of array), A057000 (signed version of second row), A362451,

Sequence in context: A365923 A089650 A085513 * A259965 A117054 A036579

Adjacent sequences: A362910 A362911 A362912 * A362914 A362915 A362916

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 09 2023

Status

approved