OFFSET
1,2
COMMENTS
Primitive positive integer solutions to Descartes' theorem for 4 kissing circles (radii r1..r4): (1/r1 + 1/r2 + 1/r3 + 1/r4) ^ 2 = 2 * ((1/r1)^2 + (1/r2)^2 + (1/r3)^2 + (1/r4)^2).
There are only a finite number of rows for each distinct r1 because r2 must be <= r1/(2/sqrt(3)-1) (otherwise r3 or r4 would be < r2), r3 must be <= 4*(1/r1+1/r2)/(1/r1-1/r2)^2 (otherwise r4 would be < r3), and there exists only 0 or 1 possible r4 given a known r1..r3.
It is observed and generally conjectured that if the largest 2 terms of a quadruple are equal, then the absolute difference between the other 2 terms is 1 or 2 times a cube.
Regular quadruples include {k, k+1, A069074(k-1), A069074(k-1)} for every positive k, {k, k+2, A000447(k)*12, A000447(k)*12} for every odd positive k, and {k, k+8, A061550((k-2)/4), A061550((k-2)/4)} for every positive k = 2 mod 4.
Some pairs of quadruples contain coincident triples; the smallest are {2, 3, 120, 120} and {3, 5, 120, 120}. It is unknown whether arbitrarily large chains of such pairs can exist (e.g. {a, b, c, d} -> {b, c, d, e} -> {c, d, e, f} -> etc.).
LINKS
Charles L. Hohn, Table of n, a(n) for n = 1..4000
EXAMPLE
Quadruples of circle radii r1..r4:
r1 r2 r3 r4
1 2 24 24
1 3 12 12
2 3 120 120
2 10 15 15
3 4 336 336
3 5 120 120
3 8 22 264
4 5 720 720
4 11 33 132
5 6 1320 1320
...
PROG
(PARI) my(r1, r2, r3, r4, a, b, c, t); for(r1=1, 100, for(r2=r1+1, ceil(r1/(2/sqrt(3)-1)), a=1/r1; b=1/r2; for(r3=r2, floor(4*(a+b)/(a-b)^2), a=r2*r3; b=r1*r3; c=r1*r2; t=a*b+a*c+b*c; if(!issquare(t), next); t=a+b+c-2*sqrtint(t); if(t<=0, next); r4=r1*r2*r3/t; if(type(r4)!="t_INT" || gcd([r1, r2, r3, r4])>1, next); print([r1, r2, r3, r4]))))
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Charles L. Hohn, Nov 11 2025
STATUS
approved