cp's OEIS Frontend

A379663 a(n) is the number of integer-sided triangles whose sides are in geometric progression with smallest side n.

%I A379663 #6 Jan 17 2025 19:38:25
%S A379663 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,3,1,2,1,2,1,1,1,2,4,1,2,2,1,1,1,3,1,1,
%T A379663 1,4,1,1,1,2,1,1,1,2,2,1,1,3,5,4,1,2,1,2,1,2,1,1,1,2,1,1,2,5,1,1,1,2,
%U A379663 1,1,1,4,1,1,4,2,1,1,1,3,6,1,1,2,1,1,1,2
%N A379663 a(n) is the number of integer-sided triangles whose sides are in geometric progression with smallest side n.
%C A379663 The integer sides of the triangles are n, n*r, n*r^2 with rational r >= 1. From the triangle inequality n + n*r >= n*r^2 follows r <= (1 + sqrt(5))/2 (golden ratio). Therefore 1 <= r = c/d < (1 + sqrt(5))/2, where c and d are coprimes and d^2 divides n.
%H A379663 Felix Huber, <a href="/A379663/b379663.txt">Table of n, a(n) for n = 1..10000</a>
%H A379663 Felix Huber, <a href="/A379663/a379663.txt">Triangles for a given n</a>
%H A379663 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>
%H A379663 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_progression">Geometric Progression</a>
%H A379663 Wikipedia, <a href="https://en.wikipedia.org/wiki/Triangle_inequality">Triangle Inequality</a>
%F A379663 a(n) = A060143(A000188(n)) + 1.
%e A379663 The a(18) = 2 integer-sided triangles whose sides form a geometric sequence are [18, 18, 18] with r = 1, [18, 24, 32] with r = 4/3.
%e A379663 The a(25) = 4 integer-sided triangles whose sides form a geometric sequence are [25, 25, 25] with r = 1, [25, 30, 36] with r = 6/5, [25, 35, 49] with r = 7/5, [25, 40, 64] with r = 8/5.
%e A379663 The a(36) = 4 integer-sided triangles whose sides form a geometric sequence are [36, 36, 36] with r = 1, [36, 54, 81] with r = 3/2, [36, 48, 64] with r = 4/3, [36, 42, 49] with r = 7/6.
%e A379663 See also the linked Maple program "Triangles for a given n".
%p A379663 A379663:=n->floor(2*expand(NumberTheory:-LargestNthPower(n,2))/(1+sqrt(5)))+1;
%p A379663 seq(A379663(n),n=1..88);
%Y A379663 Cf. A000188, A000201, A008833, A027750, A046951, A057918, A060143, A366398, A370599, A376900, A379033, A379340, A379341, A379705.
%K A379663 nonn
%O A379663 1,4
%A A379663 _Felix Huber_, Jan 07 2025