A177121 - OEIS

%I #18 Jul 28 2019 21:03:24

%S 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,

%T 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,

%U -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

%N 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.

%C 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.

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

%H Antti Karttunen, <a href="/A177121/b177121.txt">Table of n, a(n) for n = 1..65703 (the first 362 antidiagonals of array)</a>

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

%e Table begins:

%e   1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...

%e   1, -1,  1, -1,  1, -1,  1, -1,  1, -1, ...

%e   1,  1, -2,  1,  1, -2,  1,  1, -2,  1, ...

%e   1,  1,  1, -3,  1,  1,  1, -3,  1,  1, ...

%e   1,  1,  1,  1, -4,  1,  1,  1,  1, -4, ...

%e   1,  1,  1,  1,  1, -5,  1,  1,  1,  1, ...

%e   1,  1,  1,  1,  1,  1, -6,  1,  1,  1, ...

%e   1,  1,  1,  1,  1,  1,  1, -7,  1,  1, ...

%e   1,  1,  1,  1,  1,  1,  1,  1, -8,  1, ...

%e   1,  1,  1,  1,  1,  1,  1,  1,  1, -9, ...

%e   ...

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

%o (PARI)

%o up_to = 65703; \\ = binomial(362+1,2)

%o A177121sq(row,col) = if(1==row,1,if(!(col%row),(1-row),1));

%o 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); };

%o v177121 = A177121list(up_to);

%o A177121(n) = v177121[n]; \\ _Antti Karttunen_, Sep 25 2018

%Y Cf. A051731, A054535, A077049, A176079.

%K sign,tabl,look

%O 1,13

%A _Mats Granvik_ and _Gary W. Adamson_, May 03 2010