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A218856
Real part of the arithmetic derivative for the triangle of Gaussian integers z = r + i*I.
4
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
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.
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
Sequence in context: A295664 A250213 A033794 * A101665 A229951 A165034
KEYWORD
sign,tabf
AUTHOR
T. D. Noe, Nov 12 2012
STATUS
approved