A049615 Array T by antidiagonals; T(i,j) = number of lattice points (x,y) hidden from (i,j), where 0<=x<=i, 0<=y<=j; (x,y) is hidden if there is a lattice point (h,k) collinear with and between (x,y) and (i,j).

0, 0, 0, 1, 0, 1, 2, 1, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 4, 3, 4, 5, 4, 6, 6, 6, 4, 5, 6, 5, 7, 8, 8, 7, 5, 6, 7, 6, 9, 9, 11, 9, 9, 6, 7, 8, 7, 10, 12, 12, 12, 12, 10, 7, 8, 9, 8, 12, 13, 16, 14, 16, 13, 12, 8, 9, 10, 9, 13, 15, 17, 18, 18, 17, 15, 13, 9, 10

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

From Robert Israel, Jun 25 2015: (Start)

T(i,j) = number of (x,y) with 1 <= x <= i, 1 <= y <= j and gcd(x,y) > 1.

T(n,n) - T(n-1,n) = A062830(n) for x >= 2.

T(m+1,n+1) - T(m+1,n) - T(m,n+1) + T(m,n) = 1 if gcd(m+1,n+1) > 1, 0 otherwise. (End)

Examples

			Antidiagonals (each starting on row 0):
  {0};
  {0,0};
  {1,0,1};
  ...
Array begins:
  0  0  1  2  3  4  5
  0  0  1  2  3  4  5
  1  1  3  4  6  7  9
  2  2  4  6  8  9 12
  3  3  6  8 11 12 16
  4  4  7  9 12 14 18
  5  5  9 12 16 18 23
  ...
		

Links

• Ivan Neretin, Table of n, a(n) for n = 0..5049

Crossrefs

Cf. A032766, A062830.

Programs

Maple

N := 20: # to get the first N*(N+1)/2 terms
T:= Array(1..N+1,1..N+1):
B:= Array(1..N+1,1..N+1, (i,j) -> `if`(igcd(i-1,j-1)>1,1,0)):
T[1,1..N+1]:= Statistics:-CumulativeSum(B[1,1..N+1]):
for i from 2 to N+1 do
   T[i,1..N+1]:= T[i-1,1..N+1] + Statistics:-CumulativeSum(B[i,1..N+1])
od:
seq(seq(round(T[i+1,t-i+1]),i=0..t),t=0..N); # Robert Israel, Jun 25 2015
# alternative program R. J. Mathar, Oct 26 2015
A049615 := proc(n,k)
    local a,x,y;
    a := 0 ;
    for x from 0 to n do
    for y from 0 to k do
        if igcd(x,y) > 1 then
            a := a+1 ;
        end if;
    end do:
    end do:
    a;
end proc:
seq(seq(A049615(d-k,k),k=0..d),d=0..10) ;

Mathematica

Table[Length[Select[Flatten[Table[{x, y}, {x, 0, n - k}, {y, 0, k}], 1], GCD @@ # > 1 &]], {n, 0, 11}, {k, 0, n}] // Flatten (* Ivan Neretin, Jun 25 2015 *)

PARI

T(n,k) = sum(i=0, n, sum(j=0, k, gcd(i,j)>1));
tabl(7, 7, n, k, T(n-1, k-1)) \\ Michel Marcus, Aug 06 2021