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A339856 Primitive triples for integer-sided triangles whose sides a < b < c form a geometric progression. 5
4, 6, 9, 9, 12, 16, 16, 20, 25, 25, 30, 36, 25, 35, 49, 25, 40, 64, 36, 42, 49, 49, 56, 64, 49, 63, 81, 49, 70, 100, 49, 77, 121, 64, 72, 81, 64, 88, 121, 81, 90, 100, 81, 99, 121, 81, 117, 169, 81, 126, 196, 100, 110, 121, 100, 130, 169, 121, 132, 144, 121, 143, 169 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
These triangles are called "geometric triangles" in Project Euler problem 370 (see link).
The triples are displayed in increasing lexicographic order (a, b, c).
Equivalently: triples of integer-sided triangles such that b^2 = a*c with a < c and gcd(a, c) = 1.
When a < b < c are in geometric progression with b = a*q, c = b*q, q is the constant, then 1 < q < (1+sqrt(5))/2 = phi = A001622 = 1.6180... (this bound is used in Maple code).
For each triple (a, b, c), there exists (r, s), 0 < r < s such that a = r^2, b = r*s, c = s^2, q = s/r.
Angle C < 90 degrees if 1 < q < sqrt(phi) and angle C > 90 degrees if sqrt(phi) < q < phi with sqrt(phi) = A139339 = 1.2720...
For k >= 2, each triple (a, b, c) of the form (k^2, k*(k+1), (k+1)^2) is (A008133(3k+1), A008133(3k+2), A008133(3k+3)).
Three geometrical properties about these triangles:
1) The sinus satisfy sin^2(B) = sin(A) * sin(C) with sin(A) < sin(B) < sin(C) that form a geometric progression.
2) The heights satisfy h_b^2 = h_a * h_c with h_c < h_b < h_a that form a geometric progression.
3) b^2 = 2 * R * h_b, with R = circumradius of the triangle ABC.
LINKS
EXAMPLE
The smallest such triangle is (4, 6, 9) with 4*9 = 6^2.
There exist four triangles with small side = 49 corresponding to triples (49, 56, 64), (49, 63, 81), (49, 70, 100) and (49, 77, 121).
The table begins:
4, 6, 9;
9, 12, 16;
16, 20, 25;
25, 30, 36;
25, 35, 49;
25, 40, 64;
36, 42, 49;
...
MAPLE
for a from 1 to 300 do
for b from a+1 to floor((1+sqrt(5))/2 * a) do
for c from b+1 to floor((1+sqrt(5))/2 * b) do
k:=a*c;
if k=b^2 and igcd(a, b, c)=1 then print(a, b, c); end if;
end do;
end do;
end do;
PROG
(PARI) lista(nn) = {my(phi = (1+sqrt(5))/2); for (a=1, nn, for (b=a+1, floor(a*phi), for (c=b+1, floor(b*phi), if ((a*c == b^2) && (gcd([a, b, c])==1), print([a, b, c])); ); ); ); } \\ Michel Marcus, Dec 25 2020
(PARI) upto(n) = my(res=List(), phi = (sqrt(5)+1) / 2); for(i = 2, sqrtint(n), for(j = i+1, (i*phi)\1, if(gcd(i, j)==1, listput(res, [i^2, i*j, j^2])))); concat(Vec(res)) \\ David A. Corneth, Dec 25 2020
CROSSREFS
Cf. A339857 (smallest side), A339858 (middle side), A339859 (largest side), A339860 (perimeter).
Cf. A336755 (similar for sides in arithmetic progression).
Cf. A335893 (similar for angles in arithmetic progression).
Cf. A001622 (phi), A139339 (sqrt(phi)), A008133.
Sequence in context: A201660 A341577 A094115 * A163297 A051679 A010378
KEYWORD
nonn,easy,tabf
AUTHOR
Bernard Schott, Dec 19 2020
EXTENSIONS
Data corrected by David A. Corneth, Dec 25 2020
STATUS
approved

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Last modified March 29 06:57 EDT 2024. Contains 371265 sequences. (Running on oeis4.)