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

1,13

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

Links

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

Formula

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

Example

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, ...
  ...

Prog

(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

Crossrefs

Cf. A051731, A054535, A077049, A176079.

Sequence in context: A346087 A086251 A374381 * A329314 A092931 A147300

Adjacent sequences: A177118 A177119 A177120 * A177122 A177123 A177124

Keyword

sign,tabl,look

Author

Mats Granvik and Gary W. Adamson, May 03 2010

Status

approved