A288797 Square array a(p,q) = p^2 + q^2 - 2*p - 2*q + 2*gcd(p,q), p >= 1, q >= 1, read by antidiagonals.

0, 1, 1, 4, 4, 4, 9, 5, 5, 9, 16, 12, 12, 12, 16, 25, 17, 13, 13, 17, 25, 36, 28, 20, 24, 20, 28, 36, 49, 37, 33, 25, 25, 33, 37, 49, 64, 52, 40, 36, 40, 36, 40, 52, 64, 81, 65, 53, 45, 41, 41, 45, 53, 65, 81, 100, 84, 72, 64, 52, 60, 52, 64, 72, 84, 100

Offset: 1

Author

Luc Rousseau, Jun 16 2017

Keywords

• tabl
• nonn
• easy

Comments

In the Cartesian plane, let r(p,q) denote the rotation with center origin and angle associated to slope p/q (p: number of units upwards, q: number of units towards the right).

Let R(p,q) be the square of area p^2 + q^2 = R^2, with vertices (0,0), (0,R), (R,R), (R,0).

The natural unit squares (i.e., the (a,a+1) X (b,b+1) Cartesian products, with a and b integers) are transformed by r(p,q) into rotated unit squares.

a(p,q) is the number of rotated unit squares that fully land inside R(p,q).

Examples

			Table begins:
    0   1   4   9  16  25 ...
    1   4   5  12  17  28 ...
    4   5  12  13  20  33 ...
    9  12  13  24  25  36 ...
   16  17  20  25  40  41 ...
   25  28  33  36  41  60 ...
  ... ... ... ... ... ... ...
		

Links

• Luc Rousseau, Illustration for p=6, q=4

Crossrefs

Cf. A000217, A000290, A000566, A001844, A003989, A051624, A080827, A099392, A143101.

Programs

Mathematica

A[p_, q_] := p^2 + q^2 - 2*p - 2*q + 2*GCD[p, q];
(* or, checking without the formula: *)
okQ[{a_, b_}, p_, q_] := Module[{r2 = p^2 + q^2}, 0 <= a*q - b*p <= r2 && 0 <= a*p + b*q <= r2 && 0 <= a*q - b*p + q <= r2 && 0 <= a*p + b*q + p <= r2 && 0 <= a*q - (b + 1)*p <= r2 && 0 <= a*p + b*q + q <= r2 && 0 <= (a + 1)*q - (b + 1)*p <= r2 && 0 <= a*p + b*q + p + q <= r2];
A[p_, q_] := Module[{r}, r = Reduce[okQ[{a, b}, p, q], {a, b}, Integers]; If[r[[0]] === And, 1, Length[r]]];
Flatten[Table[A[p - q + 1, q], {p, 1, 11}, {q, 1, p}]] (* Jean-François Alcover, Jun 17 2017 *)

PARI

a(p,q)=p^2+q^2-2*p-2*q+2*gcd(p,q)
for(n=2,12,for(p=1,n-1,{q=n-p;print(a(p,q))}))

Formula

a(p,q) = p^2 + q^2 - 2*p - 2*q + 2*gcd(p,q).

a(p,1) = a(1,p) = (p-1)^2 = A000290(p-1).

a(p,p) = 2*p*(p-1) = 4*A000217(p-1).

a(p,p+1) = 2*p*(p-1)+1 = A001844(p-1).

a(p,p+2) = 2*p^2+2*gcd(2,p) = 2*p^2+3+(-1)^(p) = 4*A099392(p-1) = 4*A080827(p).

a(p,p+3) = 2*p^2+2*p+5+4*A079978(p) = 1+4*(1+A143101(p)).

a(p,2*p) = p*(5*p-4) = A051624(p).

a(p,3*p) = 2*p*(5*p-3) = 4*A000566(p).