The number of such strings containing 2n characters is the n th Catalan number Binomial[2n, n]/(n + 1) (as obtained from the generating function (1 - Sqrt[1 - 4x])/(2x) ).

The effective mass for massive particles increases by a factor 1/Sqrt[1 - v 2 /c 2 ] at speed v , making it take progressively more energy to increase v .

For it turns out that an angle between successive elements of about 137.5° is equivalent to a rotation by a number of turns equal to the so-called golden ratio (1+Sqrt[5])/2 ≃ 1.618 which arises in a wide variety of mathematical contexts—notably as the limiting ratio of Fibonacci numbers.

Different current methods asymptotically require slightly different numbers of steps—but all typically at least Exp[Sqrt[Log[n]]] .

The patterns are exactly repetitive only when Tan[ θ ]  u/v , where u and v are elements of a primitive Pythagorean triple (so that u , v and Sqrt[u 2 + v 2 ] are all integers, and GCD[u, v]  1 ).

After t steps, the width of the pattern shown here is about Sqrt[Log[2, 3] t] .

For rule 150R, the configuration at step t as shown in the picture on page 439 is given by (u t - v t )/Sqrt[4 + h 2 ] , where {u, v} = z /.

And if one assumes that this is a general feature then one can formally derive for any a the result 1/2 (1 - g[a t InverseFunction[g] [1 - 2x]]) where g is a function that satisfies the functional equation g[a x]  1 + (a/2) (g[x] 2 - 1) When a = 4 , g[x] is Cosh[Sqrt[2 x]] . … But for example x  a x + b and x  1/(a + b x) both give results just involving powers, while x  Sqrt[a x + b] sometimes yields trigonometric functions, as on page 915 .

[History of] exact solutions Some notable cases where closed-form analytical results have been found in terms of standard mathematical functions include: quadratic equations (~2000 BC) ( Sqrt ); cubic, quartic equations (1530s) ( x 1/n ); 2-body problem (1687) ( Cos ); catenary (1690) ( Cosh ); brachistochrone (1696) ( Sin ); spinning top (1849; 1888; 1888) ( JacobiSN ; WeierstrassP ; hyperelliptic functions); quintic equations (1858) ( EllipticTheta ); half-plane diffraction (1896) ( FresnelC ); Mie scattering (1908) ( BesselJ , BesselY , LegendreP ); Einstein equations (Schwarzschild (1916), Reissner–Nordström (1916), Kerr (1963) solutions) (rational and trigonometric functions); quantum hydrogen atom and harmonic oscillator (1927) ( LaguerreL , HermiteH ); 2D Ising model (1944) ( Sinh , EllipticK ); various Feynman diagrams (1960s-1980s) ( PolyLog ); KdV equation (1967) ( Sech etc.); Toda lattice (1967) ( Sech ); six-vertex spin model (1967) ( Sinh integrals); Calogero–Moser model (1971) ( Hypergeometric1F1 ); Yang–Mills instantons (1975) (rational functions); hard-hexagon spin model (1979) ( EllipticTheta ); additive cellular automata (1984) ( MultiplicativeOrder ); Seiberg–Witten supersymmetric theory (1994) ( Hypergeometric2F1 ).

Another way to state the Einstein equations—already discussed by David Hilbert in 1915—is as the constraint that the integral of RicciScalar Sqrt[Det[g]] (the so-called Einstein–Hilbert action) be an extremum. (An idealized soap film or other minimal surface extremizes the integral of the intrinsic volume element Sqrt[Det[g]] , without a RicciScalar factor.)