A372357 Array read by upward antidiagonals: A(n, k) = A372356(1+n,k)-2*A372356(n,k), n,k >= 1.

0, 0, 2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 1, 0, 0, 0, 3, -1, -1, 0, 0, 0, 2, -1, 3, 2, 0, 0, 0, 0, 3, 2, 0, -1, 0, 0, 0, 0, 2, 0, 0, -2, 3, 0, 0, 0, 0, 0, 0, 0, 5, 2, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 5, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 3, -1

Offset: 1

Author

Antti Karttunen, Apr 30 2024

Keywords

• sign
• tabl

Examples

			Array begins:
n\k| 1  2  3   4   5   6  7   8  9  10 11  12  13  14  15  16  17 18  19  20  21
---+-----------------------------------------------------------------------------
1  | 0, 2, 0, -1,  1, -1, 2, -1, 3,  3, 0, -2, -2, -1, -1, -1,  2, 5,  1,  1,  1,
2  | 0, 0, 0, -1, -1,  3, 0, -2, 2, -1, 0,  5,  3,  1, -1, -2, -2, 2, -1,  0, -1,
3  | 0, 0, 0,  3, -1,  2, 0,  5, 0, -1, 0,  2, -1, -1,  3,  1,  3, 0, -1, -2, -2,
4  | 0, 0, 0,  2,  3,  0, 0,  2, 0,  3, 0,  0, -1, -2,  2, -1, -1, 0,  3,  4,  1,
5  | 0, 0, 0,  0,  2,  0, 0,  0, 0,  2, 0,  0,  3,  1,  0,  2, -1, 0,  2, -2, -1,
6  | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  2, -1,  0, -1,  3, 0,  0,  3,  2,
7  | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  2,  0, -1,  2, 0,  0, -1, -1,
8  | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -1,  0,  1,  0, 0,  0, -1, -1,
9  | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -1,  0, -2,  0, 0,  0,  3,  1,
10 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  1,  0,  2,  0, 0,  0,  2, -2,
11 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -2,  0, -1,  0, 0,  0,  0,  2,
12 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  2,  0, -2,  0, 0,  0,  0, -1,
13 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -1,  0,  1,  0, 0,  0,  0, -2,
14 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -2,  0, -1,  0, 0,  0,  0,  1,
15 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  1,  0,  3,  0, 0,  0,  0, -1,
16 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -1,  0, -2,  0, 0,  0,  0,  3,
17 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  3,  0,  1,  0, 0,  0,  0, -2,
18 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -2,  0,  0,  0, 0,  0,  0,  1,
19 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  1,  0, -2,  0, 0,  0,  0,  0,
20 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  0,  0,  2,  0, 0,  0,  0, -2,
21 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -2,  0, -2,  0, 0,  0,  0,  2,
		

Crossrefs

Cf. A000523, A371094, A372282, A372354, A372356.

Programs

PARI

up_to = 105;
A000523(n) = logint(n,2);
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
A372356sq(n,k) = { my(x=A372282sq(n,k)); (A000523(A371094(x))-A000523(x)); };
A372357sq(n,k) = (A372356sq(1+n,k)-2*A372356sq(n,k));
A372357sq(n,k) = { my(x=A372282sq(n,k), y=A371094(x), z=A371094(y), xl=A000523(x), yl=A000523(y), zl=A000523(z)); ((zl-yl) - 2*(yl-xl)); };
A372357sq(n,k) = { my(x=A372282sq(n,k), y=A371094(x), z=A371094(y), xl=A000523(x), yl=A000523(y), zl=A000523(z)); (zl - 3*yl + 2*xl); };
A372357list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372357sq((a-(col-1)),col))); (v); };
v372357 = A372357list(up_to);
A372357(n) = v372357[n];