A088899 - OEIS

A088899

T(n, k) = number of ordered pairs of integers (x,y) with x^2/n^2 + y^2/k^2 = 1, 1 <= k <= n; triangular array, read by rows.

4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 12, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 12, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

T(n,k) is the number of lattice points on the circumference of an ellipse with semimajor axis = n, semiminor axis = k and center = (0,0).

LINKS

Antti Karttunen, Rows n = 1..225 of triangle, flattened

Eric Weisstein's World of Mathematics, Ellipse

FORMULA

a(n) = A088897(n) - A088898(n);

T(n,n) = A046109(n).

EXAMPLE

From Antti Karttunen, Nov 08 2018: (Start)

Triangle begins:

---------------------------------------------------------------

k= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

---------------------------------------------------------------

n= 1: 4;

n= 2: 4, 4;

n= 3: 4, 4, 4;

n= 4: 4, 4, 4, 4;

n= 5: 4, 4, 4, 4, 12;

n= 6: 4, 4, 4, 4, 4, 4;

n= 7: 4, 4, 4, 4, 4, 4, 4;

n= 8: 4, 4, 4, 4, 4, 4, 4, 4;

n= 9: 4, 4, 4, 4, 4, 4, 4, 4, 4;

n=10: 4, 4, 4, 4, 12, 4, 4, 4, 4, 12;

n=11: 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4;

n=12: 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4;

n=13: 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 12;

n=14: 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4;

n=15: 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12;

---

T(5,5) = 12 as there are following 12 solutions for pair (5,5): (5, 0), (4, 3), (3, 4), (0, 5), (-3, 4), (-4, 3), (-5, 0), (-4, -3), (-3, -4), (0, -5), (3, -4), (4, -3).

T(15,10) = 12, as there are following 12 solutions for pair (15,10): (-15,0), (-12,-6), (-12,6), (-9,-8), (-9,8), (0,-10), (0,10), (9,-8), (9,8), (12,-6), (12,6), (15,0).

(End)

MATHEMATICA

T[n_, k_] := Reduce[x^2/n^2 + y^2/k^2 == 1, {x, y}, Integers] // Length;

Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 27 2021 *)

PROG

(PARI)

up_to = 105;

A088899tr(n, k) = { my(s=0, t=(n^2)*(k^2)); for(x=-n, n, for(y=-k, k, if((x*x*k*k)+(y*y*n*n) == t, s++))); (s); };

A088899list(up_to) = { my(v = vector(up_to), i=0); for(n=1, oo, for(k=1, n, if(i++ > up_to, return(v)); v[i] = A088899tr(n, k))); (v); };

v088899 = A088899list(up_to);

A088899(n) = v088899[n]; \\ Antti Karttunen, Nov 07 2018

CROSSREFS