A088898 -id:A088898 - cp's OEIS frontend

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

Reinhard Zumkeller, Oct 21 2003

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).

			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)

Antti Karttunen, Rows n = 1..225 of triangle, flattened
Eric Weisstein's World of Mathematics, Ellipse

Cf. A046109, A088897, A088898.

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 *)

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

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

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

A088897 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.

Original entry on oeis.org

5, 7, 13, 9, 19, 29, 11, 25, 35, 49, 13, 31, 45, 63, 81, 15, 37, 55, 73, 91, 113, 17, 43, 65, 87, 109, 131, 149, 19, 45, 71, 97, 123, 145, 175, 197, 21, 51, 81, 107, 141, 163, 197, 223, 253, 23, 57, 91, 121, 159, 185, 219, 245, 279, 317, 25, 63, 101, 135, 169

Offset: 1

Reinhard Zumkeller, Oct 21 2003

T(n,k) = number of lattice points covered by an ellipse with semimajor axis = n, semiminor axis = k and center = (0,0).
a(n) = A088898(n) + A088899(n);
T(n,n) = A000328(n).

Eric Weisstein's World of Mathematics, Ellipse

Cf. A000328, A088898, A088899.

T[n_, k_] := Reduce[x^2/n^2 + y^2/k^2 <= 1, {x, y}, Integers] // Length;
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 27 2021 *)

cp's OEIS Frontend

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.

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