cp's OEIS Frontend

Irregular triangle read by rows (see examples). Consider an arbitrary anharmonic oscillator with Hamiltonian energy: H=(1/2)*b^2=(1/2)*(p^2+q^2) + Sum_{i=3} 2*v_i*q^i, and a stable minimum at (p,q)=(0,0). The phase space trajectory can be written in polar phase space coordinates as (q,p) = (R(x)cos(x),R(x)sin(x))=(R(Q)Q,R(Q)P). The present triangle determines a power / Fourier series of R(Q): R(Q) = b * (1 + sum b^n*T(n,m)*f(n,m) ); where the sum runs over n = 1,2,3 ... and m = 1,2,3...A000041(n). The basis functions f(n,m) are constructed from partitions of "n" listed in reverse lexicographic order. Partition n=(z_1+z_2+...z_j) becomes 2*Q^((z_1+2)+(z_2+2)+...(z_j+2))*v_{z_1+2}*v_{z_2+2}*...*v_{z_j+2} (see examples). This sequence transforms into A273506/A273507 by setting v_i=0 for odd i, v_i:=(-1)^(i/2-1)/2/(i!) otherwise, and (1/2)*b^2 = 2*k. For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

The phase space trajectory A276738 has phase space angular velocity A276814 and differential time dependence A276815. We calculate the period K = Int dt over the range [2*Pi, 0], trivial to compute from A276815 using A273496. Then K/(2*Pi) = 1 + sum b^(2n)*T(n,m)*f'(n,m); where the sum runs over n = 1, 2, 3 ... and m = 1, 2, 3, ... A000041(2n), and f'(n,m) = f(2n,m) of A276738 with Q=1/2. Choosing one point from the infinite dimensional coefficient space--v_i=0 for odd i, v_i=(-1)^(i/2-1)/2/(i!) otherwise--setting b^2 = 4*k, and summing over the entire table obtains the EllipticK expansion 2*A038534/A038533. For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

Irregular triangle read by rows ( see examples ). The phase space trajectory of A276738 has one time dependent variable, the phase space angle "x" defined as Tan[x]=p/q. Then dx/dt = cos[x]^2* d/dt(p/q), which can be written as a function of Q=cos[x] by application of the classical equations of motion d/dt(p,q) = ( -d/dq H, d/dp H ), with H the anharmonic oscillator Hamiltonian. Substituting the result of A276738 and expanding in powers of b, we obtain dx/dt = -1 + sum b^n*T(n,m)*f(n,m); where the sum runs over n=1,2,3... and m = 1,2,3, ... A000041(n). The basis functions f(n,m) are the same as in A276738. Observe the limit where Q --> 0, dx/dt --> -1, the harmonic oscillator value. Similarly if v_i --> 0 then dx/dt --> -1.

The phase space trajectory A276738 has phase space angular velocity A276814, which allows expansion of dt = dx /(dx/dt) = dx(-1 + sum b^n*T(n,m)*f(n,m)); where the sum runs over n = 1, 2, 3 ... and m = 1, 2, 3, ... A000041(n). The basis functions f(n,m) are the same as in A276738. To obtain period K, we integrate the function of Q=cos[x] over a range of [2*pi,0]. All odd powers of Q integrate to zero, so the period is an expansion in E=(1/2)*b^2 (Cf. A276816). This sequence transforms into A274076/A274078 by setting v_i=0 for odd i, v_i=(-1)^(i/2-1)/2/(i!) otherwise, and (1/2)*b^2 = 2*k. For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).