OFFSET
0,6
COMMENTS
For n > 0, the 4*n terms in row n are computed starting at (n,0) and proceeding counterclockwise (n-1,1), (n-2,2), ..., (0,n), (-1,n-1), (-2,n-2), ..., (-n,0), (-n+1,-1), (-n+2,-2), ..., (0,-n), (1,-n+1), (2,-n+2), ..., (n-1,-1). See the two-dimensional plot to see both the real and the imaginary parts. The imaginary part is in A218857. Consult A099379 for the arithmetic derivative of Gaussian integers.
LINKS
T. D. Noe, Rows n = 0..50 of triangle, flattened
T. D. Noe, 2-d plots of the derivative for n = 3..25
EXAMPLE
Triangle:
0,
0, 0, 0, 0,
2, 1, 2, 0, -2, -1, -2, 0,
1, 1, 1, 0, 0, 0, -1, -1, -1, 0, 0, 0,
8, 3, 6, 3, 8, 2, 0, -2, -8, -3, -6, -3, -8, -2, 0, 2,
3, 1, 1, 1, 1, 3, 0, 0, 0, 0, -3, -1, -1, -1, -1, -3, 0, 0, 0, 0,
8, 4, 8, 4, 8, 4, 6, 3, 2, -1, -2, -3, -8, -4, -8, -4, -8, -4, -6, -3, -2, 1, 2, 3
MATHEMATICA
di[0]=0; di[1]=0; di[ -1]=0; di[I]=0; di[ -I]=0; di[n_] := Module[{f, unt}, f=FactorInteger[n, GaussianIntegers->True]; unt=(Abs[f[[1, 1]]]==1); If[unt, f=Delete[f, 1]]; f=Transpose[f]; Plus@@(n*f[[2]]/f[[1]])]; Table[t = Join[Table[di[n - i + I*i], {i, 0, n}], Table[di[i - n + I*i], {i, n - 1, 0, -1}], Table[di[i - n - I*i], {i, 1, n}], Table[di[n - i - I*i], {i, n - 1, 1, -1}]]; Re[t], {n, 0, 6}]
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
T. D. Noe, Nov 12 2012
STATUS
approved