A218856 Real part of the arithmetic derivative for the triangle of Gaussian integers z = r + i*I.

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.

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

Cf. A099379, A099380.

Sequence in context: A295664 A250213 A033794 * A101665 A229951 A165034

Adjacent sequences: A218853 A218854 A218855 * A218857 A218858 A218859

Keyword

sign,tabf

Author

T. D. Noe, Nov 12 2012

Status

approved