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

1,3

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.

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

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

Example

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
  ...

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 *)

Crossrefs

Cf. A000005, A036262, A051950, A361897.

Sequence in context: A071412 A080884 A354452 * A298731 A321102 A368199

Adjacent sequences: A362447 A362448 A362449 * A362451 A362452 A362453

Keyword

nonn,tabl

Author

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

Status

approved