A049615 - OEIS

%I #29 Aug 29 2024 16:00:08

%S 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 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 13,12,8,9,10,9,13,15,17,18,18,17,15,13,9,10

%N 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 From _Robert Israel_, Jun 25 2015: (Start)

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

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

%C 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 Ivan Neretin, <a href="/A049615/b049615.txt">Table of n, a(n) for n = 0..5049</a>

%e Antidiagonals (each starting on row 0):

%e   {0};

%e   {0,0};

%e   {1,0,1};

%e   ...

%e Array begins:

%e   0  0  1  2  3  4  5

%e   0  0  1  2  3  4  5

%e   1  1  3  4  6  7  9

%e   2  2  4  6  8  9 12

%e   3  3  6  8 11 12 16

%e   4  4  7  9 12 14 18

%e   5  5  9 12 16 18 23

%e   ...

%p N := 20: # to get the first N*(N+1)/2 terms

%p T:= Array(1..N+1,1..N+1):

%p B:= Array(1..N+1,1..N+1, (i,j) -> `if`(igcd(i-1,j-1)>1,1,0)):

%p T[1,1..N+1]:= Statistics:-CumulativeSum(B[1,1..N+1]):

%p for i from 2 to N+1 do

%p    T[i,1..N+1]:= T[i-1,1..N+1] + Statistics:-CumulativeSum(B[i,1..N+1])

%p od:

%p seq(seq(round(T[i+1,t-i+1]),i=0..t),t=0..N); # _Robert Israel_, Jun 25 2015

%p # alternative program _R. J. Mathar_, Oct 26 2015

%p A049615 := proc(n,k)

%p     local a,x,y;

%p     a := 0 ;

%p     for x from 0 to n do

%p     for y from 0 to k do

%p         if igcd(x,y) > 1 then

%p             a := a+1 ;

%p         end if;

%p     end do:

%p     end do:

%p     a;

%p end proc:

%p seq(seq(A049615(d-k,k),k=0..d),d=0..10) ;

%t 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 (PARI) T(n,k) = sum(i=0, n, sum(j=0, k, gcd(i,j)>1));

%o tabl(7, 7, n, k, T(n-1, k-1)) \\ _Michel Marcus_, Aug 06 2021

%Y Cf. A032766, A062830.

%K nonn,tabl

%O 0,7

%A _Clark Kimberling_