A057918 -id:A057918 - cp's OEIS frontend

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

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, 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, 1, 1, 1, 4, 1, 1, 4, 2, 1, 1, 1, 3, 6, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Felix Huber, Jan 07 2025

nonn

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.

			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.
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.
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.
See also the linked Maple program "Triangles for a given n".

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Triangles for a given n
Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Geometric Progression
Wikipedia, Triangle Inequality

Cf. A000188, A000201, A008833, A027750, A046951, A057918, A060143, A366398, A370599, A376900, A379033, A379340, A379341, A379705.

A379663:=n->floor(2*expand(NumberTheory:-LargestNthPower(n,2))/(1+sqrt(5)))+1;
seq(A379663(n),n=1..88);

a(n) = A060143(A000188(n)) + 1.

A379705 a(n) is the least integer k > n such that integers p, q exist for which n, p, k are in arithmetic and n, q, k are in geometric progression.

Original entry on oeis.org

9, 8, 27, 16, 45, 24, 63, 18, 25, 40, 99, 48, 117, 56, 135, 36, 153, 32, 171, 80, 189, 88, 207, 54, 49, 104, 75, 112, 261, 120, 279, 50, 297, 136, 315, 64, 333, 152, 351, 90, 369, 168, 387, 176, 125, 184, 423, 108, 81, 72, 459, 208, 477, 96, 495, 126, 513, 232

Offset: 1

Felix Huber, Jan 07 2025

nonn

			a(9) = 25 because 9, 17, 25 are in arithmetic progression (common difference = 8) and 9, +-15, 25 are in geometric progression (common ratio = +-5/3) and there is no other integer k with 9 < k < 25 such that integers p and q exist for which 9, p, k are in arithmetic and 9, q, k are in geometric progression.

Felix Huber, Table of n, a(n) for n = 1..10000
Wikipedia, Arithmetic Progression
Wikipedia, Geometric Progression

Cf. A000188, A008833, A057918, A072905, A379663.

A379705:=proc(n)
   local d;
   d:=expand(NumberTheory:-LargestNthPower(n,2));
   if is(n*(1+(d+1)^2/d^2),even) then
      n*(d+1)^2/d^2
   else
      n*(d+2)^2/d^2
   fi;
end proc;
seq(A379705(n),n=1..58);

a(n) = n/A008833(n)*(A000188(n) + k)^2, where k = 1 if n*(1+(A000188(n)+1)^2/A008833(n)) is even or k = 2 else.

a(n) = A072905(n) if n*(1+(A000188(n)+1)^2/A008833(n)) is even.

cp's OEIS Frontend

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

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