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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).

%I A049615 #29 Aug 29 2024 16:00:08
%S A049615 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,
%T A049615 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,
%U A049615 13,12,8,9,10,9,13,15,17,18,18,17,15,13,9,10
%N 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).
%C A049615 From _Robert Israel_, Jun 25 2015: (Start)
%C A049615 T(i,j) = number of (x,y) with 1 <= x <= i, 1 <= y <= j and gcd(x,y) > 1.
%C A049615 T(n,n) - T(n-1,n) = A062830(n) for x >= 2.
%C A049615 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)
%H A049615 Ivan Neretin, <a href="/A049615/b049615.txt">Table of n, a(n) for n = 0..5049</a>
%e A049615 Antidiagonals (each starting on row 0):
%e A049615   {0};
%e A049615   {0,0};
%e A049615   {1,0,1};
%e A049615   ...
%e A049615 Array begins:
%e A049615   0  0  1  2  3  4  5
%e A049615   0  0  1  2  3  4  5
%e A049615   1  1  3  4  6  7  9
%e A049615   2  2  4  6  8  9 12
%e A049615   3  3  6  8 11 12 16
%e A049615   4  4  7  9 12 14 18
%e A049615   5  5  9 12 16 18 23
%e A049615   ...
%p A049615 N := 20: # to get the first N*(N+1)/2 terms
%p A049615 T:= Array(1..N+1,1..N+1):
%p A049615 B:= Array(1..N+1,1..N+1, (i,j) -> `if`(igcd(i-1,j-1)>1,1,0)):
%p A049615 T[1,1..N+1]:= Statistics:-CumulativeSum(B[1,1..N+1]):
%p A049615 for i from 2 to N+1 do
%p A049615    T[i,1..N+1]:= T[i-1,1..N+1] + Statistics:-CumulativeSum(B[i,1..N+1])
%p A049615 od:
%p A049615 seq(seq(round(T[i+1,t-i+1]),i=0..t),t=0..N); # _Robert Israel_, Jun 25 2015
%p A049615 # alternative program _R. J. Mathar_, Oct 26 2015
%p A049615 A049615 := proc(n,k)
%p A049615     local a,x,y;
%p A049615     a := 0 ;
%p A049615     for x from 0 to n do
%p A049615     for y from 0 to k do
%p A049615         if igcd(x,y) > 1 then
%p A049615             a := a+1 ;
%p A049615         end if;
%p A049615     end do:
%p A049615     end do:
%p A049615     a;
%p A049615 end proc:
%p A049615 seq(seq(A049615(d-k,k),k=0..d),d=0..10) ;
%t A049615 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 *)
%o A049615 (PARI) T(n,k) = sum(i=0, n, sum(j=0, k, gcd(i,j)>1));
%o A049615 tabl(7, 7, n, k, T(n-1, k-1)) \\ _Michel Marcus_, Aug 06 2021
%Y A049615 Cf. A032766, A062830.
%K A049615 nonn,tabl
%O A049615 0,7
%A A049615 _Clark Kimberling_