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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.

%I A362913 #27 Sep 27 2023 15:03:26
%S A362913 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,
%T A362913 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,
%U A362913 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
%N 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.
%C A362913 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.
%H A362913 Paolo Xausa, <a href="/A362913/b362913.txt">Table of n, a(n) for n = 1..11325</a> (antidiagonals 1..150 of the array, flattened)
%H A362913 N. J. A. Sloane, <a href="/A362451/a362451_2.txt">Maple code for Gilbreath transform and related arrays</a>
%H A362913 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: <a href="https://vimeo.com/866583736?share=copy">Video</a>, <a href="http://neilsloane.com/doc/EMSep2023.pdf">Slides</a>, <a href="http://neilsloane.com/doc/EMSep2023.Updates.txt">Updates</a>. (Mentions this sequence.)
%e A362913 The array begins:
%e A362913   1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, ...
%e A362913   0, 1, 0, 2, 2, 4, 2, 2, 2, 6, 6, 8, 6, 2, 0, ...
%e A362913   1, 1, 2, 0, 2, 2, 0, 0, 4, 0, 2, 2, 4, 2, ...
%e A362913   0, 1, 2, 2, 0, 2, 0, 4, 4, 2, 0, 2, 2, ...
%e A362913   1, 1, 0, 2, 2, 2, 4, 0, 2, 2, 2, 0, ...
%e A362913   0, 1, 2, 0, 0, 2, 4, 2, 0, 0, 2, ...
%e A362913   1, 1, 2, 0, 2, 2, 2, 2, 0, 2, ...
%e A362913   0, 1, 2, 2, 0, 0, 0, 2, 2, ...
%e A362913   1, 1, 0, 2, 0, 0, 2, 0, ...
%e A362913   ...
%e A362913 The first few antidiagonals are:
%e A362913   1
%e A362913   0, 1
%e A362913   1, 1, 2
%e A362913   0, 1, 0, 2
%e A362913   1, 1, 2, 2, 4
%e A362913   0, 1, 2, 0, 2, 2
%e A362913   1, 1, 0, 2, 2, 4, 6
%e A362913   0, 1, 2, 2, 0, 2, 2, 4
%e A362913   1, 1, 2, 0, 2, 2, 0, 2, 6
%e A362913   0, 1, 2, 0, 0, 2, 0, 0, 2, 4
%e A362913   ...
%t A362913 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 *)
%Y A362913 Cf. A000010 (top row of array), A057000 (signed version of second row), A362451,
%K A362913 nonn,tabl
%O A362913 1,6
%A A362913 _N. J. A. Sloane_, May 09 2023