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

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

Cf. A046109, A088897, A088898.

Sequence in context: A032564 A141248 A273339 * A258199 A290205 A066014

Adjacent sequences: A088896 A088897 A088898 * A088900 A088901 A088902

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Oct 21 2003

Status

approved