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.

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, 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, 1, 1, 1, 0, 0, 0, 1, 2, 4, 6, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 0, 4, 2

Offset: 1

Author

N. J. A. Sloane, Apr 30 2023, following a suggestion from Wayman Eduardo Luy and Robert G. Wilson v, Mar 28 2023

Keywords

• nonn
• tabl

Comments

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

Wayman Eduardo Luy and Robert G. Wilson v conjecture (see A361897) that the leading terms in the array are always 0 or 1.

Examples

			The array begins:
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  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 ...
  ...
The first few antidiagonals are
  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 2 4
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)
• Index entries for sequences related to Gilbreath conjecture and transform

Crossrefs

Cf. A000005, A036262, A051950, A361897.

Programs

Maple

A362450 := proc(n,k)
    option remember ;
    if n = 1 then
        numtheory[tau](k) ;
    else
        abs( procname(n-1,k+1)-procname(n-1,k)) ;
    end if;
end proc:
seq(seq(A362450(d-k,k),k=1..d-1),d=2..14) ; # R. J. Mathar, May 05 2023

Mathematica

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 *)

Formula

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