cp's OEIS Frontend

"The square root of 3, the 2nd number, after root 2, to be proved irrational, by Theodorus."

Length of a diagonal between any vertex of the unit cube and the one corresponding (opposite) vertex not part of the three faces meeting at the original vertex. (Diagonal is hypotenuse of a triangle with sides 1 and sqrt(2)). Hence the diameter of the sphere circumscribed around the unit cube; the ratio of the diameter of any sphere to the edge length of its inscribed cube. - Rick L. Shepherd, Jun 09 2005

The square root of 3 is the length of the minimal Y-shaped (symmetrical) network linking three points unit distance apart. - Lekraj Beedassy, Apr 12 2006

Continued fraction expansion is 1 followed by {1, 2} repeated. - Harry J. Smith, Jun 01 2009

Also, tan(Pi/3) = 2 sin(Pi/3). - M. F. Hasler, Oct 27 2011

Surface of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012

This is the case n=6 of Gamma(1/n)*Gamma((n-1)/n)/(Gamma(2/n)*Gamma((n-2)/n)) = 2*cos(Pi/n), therefore sqrt(3) = A175379*A203145/(A073005*A073006). - Bruno Berselli, Dec 13 2012

Ratio of base length to leg length in the isosceles "vampire" triangle, that is, the only isosceles triangle without reflection triangle. The product of cosines of the internal angles of a triangle with sides 1, 1 and sqrt(3) and all similar triangles is -3/8. Hence its reflection triangle is degenerate. See the link below. - Martin Janecke, May 09 2013

Half of the surface of regular octahedron with unit edge (A010469), and one fifth that of a regular icosahedron with unit edge (i.e., 2*A010527). - Stanislav Sykora, Nov 30 2013

Diameter of a sphere whose surface area equals 3*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Nov 11 2018

Sometimes called Theodorus's constant, after the ancient Greek mathematician Theodorus of Cyrene (5th century BC). - Amiram Eldar, Apr 02 2022

For any triangle ABC, cotan(A) + cotan(B) + cotan(C) >= sqrt(3); equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 13 2022

For a brief description of the modified Engel expansion of a real number see A220335.

Let p >= 2 be an integer and set Q(p) = (p - 1)*sqrt(p^2 - 1) - (p^2 - p - 1), so Q(4) = 3*sqrt(15) - 11. Iterating the identity Q(p) = 1/2 + 1/(2*(p+1)) + 1/(2*(p+1)*(2*p)) + 1/(2*(p+1)*(2*p))*Q(2*p^2-1) leads to a representation for Q(p) as an infinite series of unit fractions. The sequence of denominators of these unit fractions can be used to find the modified Engel expansion of Q(p). For further details see the Bala link. The present sequence is the case p = 4. For other cases see A220335 (p = 2), A220336 (p = 3) and A220338 (p = 5).

For a brief description of the modified Engel expansion of a real number see A220335.

Let p >= 2 be an integer and set Q(p) = (p - 1)*sqrt(p^2 - 1) - (p^2 - p - 1), so Q(3) = 4*sqrt(2) - 5. Iterating the identity Q(p) = 1/2 + 1/(2*(p+1)) + 1/(2*(p+1)*(2*p)) + 1/(2*(p+1)*(2*p))*Q(2*p^2-1) leads to a representation for Q(p) as an infinite series of unit fractions. The sequence of denominators of these unit fractions can be used to find the modified Engel expansion of Q(p). For further details see the Bala link. The present sequence is the case p = 3. For other cases see A220335 (p = 2), A220337 (p = 4) and A220338 (p = 5).

For a brief description of the modified Engel expansion of a real number see A220335.

Let p >= 2 be an integer and set Q(p) = (p - 1)*sqrt(p^2 - 1) - (p^2 - p - 1), so Q(5) = 8*sqrt(6) - 19. Iterating the identity Q(p) = 1/2 + 1/(2*(p+1)) + 1/(2*(p+1)*(2*p)) + 1/(2*(p+1)*(2*p))*Q(2*p^2-1) leads to a representation for Q(p) as an infinite series of unit fractions. The sequence of denominators of these unit fractions can be used to find the modified Engel expansion of Q(p). For further details see the Bala link. The present sequence is the case p = 5. For other cases see A220335 (p = 2), A220336 (p = 3) and A220337 (p = 4).

The Engel expansion of a positive real number x is the unique nondecreasing sequence {e(1), e(2), e(3), ...} of positive integers such that x = 1/e(1) + 1/(e(1)*e(2)) + 1/(e(1)*e(2)*e(3)) + .... The terms in the Engel expansion of x are obtained from the iterates of the mapping g(x) := x*(1 + floor(1/x)) - 1 by means of the formula e(n) = 1 + floor(1/g^(n)(x)).

Let h(x) = floor(1/x)*g(x). This is called the harmonic sawtooth map by Crowley. Then the modified Engel expansion of a real number 0 < x <= 1 is a sequence {a(1), a(2), a(3), ...} of positive integers such that x = 1/a(1) + 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) + ... whose terms are obtained from the iterates of the harmonic sawtooth map h(x) by the formulas a(1) = 1 + floor(1/x) and, for n >= 2, a(n) = floor(1/h^(n-2)(x))*{1 + floor(1/h^(n-1)(x))}. Here h^(n)(x) = h(h^(n-1)(x)) denotes the n-th iterate of the map h(x), with the convention that h^(0)(x) = x. For further details see the Bala link.

When the real number x > 1 with, say, floor(x) = m, the modified Engel expansion of x is found by first calculating the modified Engel expansion of x - m and then prepending a sequence of m 1's to this.

See A220393 for a definition of the modified Engel expansion of a positive real number. For further details see the Bala link.

See A220393 for a description of the modified Engel expansion of a positive real number. For further details see the Bala link.

The Engel expansion for exp(1) is the sequence of positive integers A000027.

See A220393 for the definition of the modified Engel expansion of a positive real number. For further details see the Bala link.

See A220393 for the definition of the modified Engel expansion of a positive real number. For further details see the Bala link.