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

1,4

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).

Links

Table of n, a(n) for n=1..66.

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

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).

Example

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 ...
  ... ... ... ... ... ... ...

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 *)

Prog

(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))}))

Crossrefs

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

Sequence in context: A183030 A176213 A065734 * A200600 A048761 A211547

Adjacent sequences: A288794 A288795 A288796 * A288798 A288799 A288800

Keyword

tabl,nonn,easy

Author

Luc Rousseau, Jun 16 2017

Status

approved