A177121 Square array T(n,k) read by antidiagonals up: T(n,k) = 1 if n=1; otherwise if n divides k then T(n,k) = -n+1; otherwise T(n,k) = 1.

1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -3, 1, -1, 1, 1, 1, 1, 1, 1, -2, 1, 1, 1, 1, 1, 1, -4, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -5, 1, -3, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -6, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Mats Granvik and Gary W. Adamson, May 03 2010

Keywords

• sign
• tabl
• look

Comments

Replace the first column in A077049 with any k-th column in A177121 (this array) to get a new array. Then the matrix inverse of the new array will have the k-th column of A054535 (Ramanujan sum) as its first column.

Obtained from A176079 by transposing, flipping signs, and adding a lower triangle of all -1's. - R. J. Mathar, Jul 08 2011

Examples

			Table begins:
  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1, -1,  1, -1,  1, -1,  1, -1,  1, -1, ...
  1,  1, -2,  1,  1, -2,  1,  1, -2,  1, ...
  1,  1,  1, -3,  1,  1,  1, -3,  1,  1, ...
  1,  1,  1,  1, -4,  1,  1,  1,  1, -4, ...
  1,  1,  1,  1,  1, -5,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1, -6,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1, -7,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, -8,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1,  1, -9, ...
  ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65703 (the first 362 antidiagonals of array)

Crossrefs

Cf. A051731, A054535, A077049, A176079.

Programs

Excel

=if(row()=1;1;if(mod(column();row())=0;-row()+1;1))

PARI

up_to = 65703; \\ = binomial(362+1,2)
A177121sq(row,col) = if(1==row,1,if(!(col%row),(1-row),1));
A177121list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, if(i++ > up_to, return(v)); v[i] = A177121sq((a-(col-1)),col))); (v); };
v177121 = A177121list(up_to);
A177121(n) = v177121[n]; \\ Antti Karttunen, Sep 25 2018

Formula

T(n,k) = 1 if n=1; otherwise, if n divides k then T(n,k) = -n+1; otherwise T(n,k) = 1.