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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 (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)

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

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Last modified September 20 16:50 EDT 2021. Contains 347586 sequences. (Running on oeis4.)