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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.
3
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
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
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Oct 21 2003
STATUS
approved