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

0,7

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)

Links

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

Example

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
  ...

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 *)

Prog

(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

Crossrefs

Cf. A032766, A062830.

Sequence in context: A029260 A205725 A091093 * A114919 A087917 A330334

Adjacent sequences: A049612 A049613 A049614 * A049616 A049617 A049618

Keyword

nonn,tabl

Author

Clark Kimberling

Status

approved