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A362450 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.

%I A362450 #38 Sep 25 2023 19:30:36
%S A362450 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 A362450 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 A362450 1,1,1,0,0,0,1,2,4,6,0,1,0,1,0,1,1,1,1,2,0,4,2
%N A362450 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 A362450 Analogous to the array in A036262 that arises from Gilbreath's conjecture.
%C A362450 Wayman Eduardo Luy and _Robert G. Wilson v_ conjecture (see A361897) that the leading terms in the array are always 0 or 1.
%H A362450 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 A362450 <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>
%F A362450 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 A362450 The array begins:
%e A362450   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 A362450   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 A362450   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 A362450   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 A362450   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 A362450   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 A362450   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 A362450   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 A362450   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 A362450   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 A362450   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 A362450   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 A362450   ...
%e A362450 The first few antidiagonals are
%e A362450   1
%e A362450   1 2
%e A362450   1 0 2
%e A362450   0 1 1 3
%e A362450   1 1 0 1 2
%e A362450   1 0 1 1 2 4
%e A362450   1 0 0 1 0 2 2
%e A362450   0 1 1 1 0 0 2 4
%e A362450   ...
%p A362450 A362450 := proc(n,k)
%p A362450     option remember ;
%p A362450     if n = 1 then
%p A362450         numtheory[tau](k) ;
%p A362450     else
%p A362450         abs( procname(n-1,k+1)-procname(n-1,k)) ;
%p A362450     end if;
%p A362450 end proc:
%p A362450 seq(seq(A362450(d-k,k),k=1..d-1),d=2..14) ; # _R. J. Mathar_, May 05 2023
%t A362450 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 A362450 Cf. A000005, A036262, A051950, A361897.
%K A362450 nonn,tabl
%O A362450 1,3
%A A362450 _N. J. A. Sloane_, Apr 30 2023, following a suggestion from Wayman Eduardo Luy and _Robert G. Wilson v_, Mar 28 2023