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Array of numbers read by upward antidiagonals: leading row lists tau(i), i >= 1 (cf. A000005); the following rows give absolute values of differences of previous row.
6

%I #38 Sep 25 2023 19:30:36

%S 1,1,2,1,0,2,0,1,1,3,1,1,0,1,2,1,0,1,1,2,4,1,0,0,1,0,2,2,0,1,1,1,0,0,

%T 2,4,0,0,1,0,1,1,1,1,3,0,0,0,1,1,0,1,0,1,4,0,0,0,0,1,0,0,1,1,2,2,1,1,

%U 1,1,1,0,0,0,1,2,4,6,0,1,0,1,0,1,1,1,1,2,0,4,2

%N Array of numbers read by upward antidiagonals: leading row lists tau(i), i >= 1 (cf. A000005); the following rows give absolute values of differences of previous row.

%C Analogous to the array in A036262 that arises from Gilbreath's conjecture.

%C Wayman Eduardo Luy and _Robert G. Wilson v_ conjecture (see A361897) that the leading terms in the array are always 0 or 1.

%H Paolo Xausa, <a href="/A362450/b362450.txt">Table of n, a(n) for n = 1..11325</a> (antidiagonals 1..150 of the array, flattened)

%H <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>

%F T(1,k) = A000005(k). T(n,k) = |T(n-1,k+1)-T(n-1,k)| for n>=2. - _R. J. Mathar_, May 10 2023

%e The array begins:

%e 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 4 4 6 2 8 2 6 4 4 4 9 2 4 4 ...

%e 1 0 1 1 2 2 2 1 1 2 4 4 2 0 1 3 4 4 4 2 0 2 6 5 1 0 2 4 6 6 4 2 0 0 5 7 2 0 4 ...

%e 1 1 0 1 0 0 1 0 1 2 0 2 2 1 2 1 0 0 2 2 2 4 1 4 1 2 2 2 0 2 2 2 0 5 2 5 2 4 2 ...

%e 0 1 1 1 0 1 1 1 1 2 2 0 1 1 1 1 0 2 0 0 2 3 3 3 1 0 0 2 2 0 0 2 5 3 3 3 2 2 2 ...

%e 1 0 0 1 1 0 0 0 1 0 2 1 0 0 0 1 2 2 0 2 1 0 0 2 1 0 2 0 2 0 2 3 2 0 0 1 0 0 2 ...

%e 1 0 1 0 1 0 0 1 1 2 1 1 0 0 1 1 0 2 2 1 1 0 2 1 1 2 2 2 2 2 1 1 2 0 1 1 0 2 0 ...

%e 1 1 1 1 1 0 1 0 1 1 0 1 0 1 0 1 2 0 1 0 1 2 1 0 1 0 0 0 0 1 0 1 2 1 0 1 2 2 2 ...

%e 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 ...

%e 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 2 ...

%e 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 2 0 ...

%e 0 1 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 1 0 1 2 2 1 ...

%e 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 1 1 1 1 0 1 1 ...

%e ...

%e The first few antidiagonals are

%e 1

%e 1 2

%e 1 0 2

%e 0 1 1 3

%e 1 1 0 1 2

%e 1 0 1 1 2 4

%e 1 0 0 1 0 2 2

%e 0 1 1 1 0 0 2 4

%e ...

%p A362450 := proc(n,k)

%p option remember ;

%p if n = 1 then

%p numtheory[tau](k) ;

%p else

%p abs( procname(n-1,k+1)-procname(n-1,k)) ;

%p end if;

%p end proc:

%p seq(seq(A362450(d-k,k),k=1..d-1),d=2..14) ; # _R. J. Mathar_, May 05 2023

%t A362450[dmax_]:=With[{d=Reverse[NestList[Abs[Differences[#]]&,DivisorSigma[0,Range[dmax]],dmax-1]]},Array[Diagonal[d,#]&,dmax,1-dmax]];A362450[20] (* Generates 20 antidiagonals *) (* _Paolo Xausa_, May 07 2023 *)

%Y Cf. A000005, A036262, A051950, A361897.

%K nonn,tabl

%O 1,3

%A _N. J. A. Sloane_, Apr 30 2023, following a suggestion from Wayman Eduardo Luy and _Robert G. Wilson v_, Mar 28 2023