cp's OEIS Frontend

Defined by Sum_{n>=1} 1/A005900(n) = 1/1 + 1/6 + 1/19 + 1/44 + ...

Equals 3*(gamma + Re psi(i/sqrt 2) ) = 3* Re(A001620 + psi(i*A010503)) where psi(i*A010503) = -0.1511539... + i*2.3152942... is a digamma function and i the imaginary unit.

Height of a equilateral square antiprism of edge length 1. - R. J. Mathar, Mar 06 2025

The corresponding imaginary part is in A232736.

Root of the equation -7 + 56*x^2 - 112*x^4 + 64*x^6 = 0. - Vaclav Kotesovec, Apr 04 2021

The corresponding real part is in A232737.

The maximum of a(n)/n appears to converge to sqrt(2)/2 (A010503), i.e. n*(n+1)/2 seems not more than n*sqrt(2)/2 distant from a square.

Some values don't seem to be in the sequence (checked up to n=10^7): 7,18,23,31,37,38...

Those values k are not in the sequence because the Pell-type equations x^2 - 8*y^2 = 8*k+1 and x^2 - 8*y^2 = -8*k+1 have no solutions. - Robert Israel, Apr 08 2019

a(A001108(n)) = 0, a(A229131(n)) = 1, a(A229083(n)) <= 1, a(A229133(n)) is square.

Also decimal expansion of the speed b = c/sqrt(2) in SI units (meter/second), where c = 299792458 (m/s) is the speed of light in vacuum (A003678).

A particle (or object) with speed b has the property that its relativistic momentum equals the momentum of a virtual photon whose energy equals the rest energy of the particle. Also its relativistic de Broglie wavelength equals the Compton wavelength for the particle and therefore equals the wavelength of the photon mentioned above.

More generally it appears that the speed b is a critical speed for several relativistic magnitudes of the particle. Explanation: consider a table of relativistic magnitudes in which every formula is written as the product of a dimensionless factor and a constant with the same dimensions as the relativistic magnitude. For instance, for the relativistic momentum we write the formula p = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c instead of the standard formula p = [1/(1 - v^2/c^2)^(1/2)]*m_0*v. See below:

Table 1.

----------------------------------------------------

Relativistic

magnitude Formula

----------------------------------------------------

Speed.........: v = [v/c]*c

Group velocity: g = [v/c]*c

Length........: L = [1/γ]*L_0

Momentum......: p = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c

Wavenumber....: k = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c/h

Wavelength....: W = [(c^2/v^2 - 1)^(1/2)]*h/(m_0*c)

Time interval.: t = γ*t_0

Mass..........: m = γ*m_0

Energy........: E = γ*m_0*c^2

Frequency.....: f = γ*m_0*c^2/h

Phase velocity: w = [c/v]*c

Kinetic energy: K = [γ - 1]*m_0*c^2

----------------------------------------------------

Where:

v is the speed of the object or particle.

c is the speed of light in vacuum (A003678).

h is the Planck constant (A003676).

L_0 is the length at rest of the object or the length at rest of a virtual cube which contains the particle.

m_0 is the mass at rest (for the electron see A081801, for the proton see A070059).

t_0 is the time interval at rest.

W is the relativistic de Broglie wavelength assuming that W = h/p.

γ = [1/(1 - v^2/c^2)^(1/2)] is the Lorentz factor.

Then table 1 can be unified as shown below:

Table 2. Table 3.

------------------------------------ -----------------

Relativistic

magnitude Formula Formula

------------------------------------ -----------------

Speed.........: v = sin(x) * c v = sin(x) * v’

Group velocity: g = sin(x) * c g = sin(x) * g’

Length........: L = cos(x) * L_0 L = cos(x) * L’

Momentum......: p = tan(x) * m_0*c p = tan(x) * p’

Wavenumber....: k = tan(x) * 1/W_C k = tan(x) * k’

Wavelength....: W = cot(x) * W_C W = cot(x) * W’

Time interval.: t = sec(x) * t_0 t = sec(x) * t’

Mass..........: m = sec(x) * m_0 m = sec(x) * m’

Energy........: E = sec(x) * E_0 E = sec(x) * E’

Frequency.....: f = sec(x) * E_0/h f = sec(x) * f’

Phase velocity: w = csc(x) * c w = csc(x) * w’

Kinetic energy: K = ese(x) * E_0 K = ese(x) * K’

------------------------------------ -----------------

Where:

E_0 = m_0*c^2 is the energy at rest (for the electron see A081816, for the proton see A230438).

W_C = h/(m_0*c) is the Compton wavelength for the particle (for the electron see A230436, for the proton see A230845).

ese(x) = sec(x) - 1.

Table 2 is simpler than table 1 because the relativistic factors are written as trigonometric functions of the angle x assuming that sin(x) = v/c and that 0 < x < Pi/2.

Table 3 lists the simplest formulas in which the values of the constants have been interpreted as the values of the magnitudes of a virtual photon whose energy E' = h*f' is equivalent to E_0 = m_0*c^2, the rest energy of the particle.

A visualization of the relationship between the relativistic magnitudes, the quantum constants and the trigonometric functions is obtained using the first quadrant of the trigonometric circle according to the simplest table, see below:

Table 4.

-----------------------------------

sin(x) = v/v' = g/g'

cos(x) = L/L'

tan(x) = p/p' = k/k'

cot(x) = W/W'

sec(x) = t/t' = m/m' = E/E' = f/f'

csc(x) = w/w'

ese(x) = K/K'

-----------------------------------

Finally we can write that b is a critical speed because:

If v = b, for instance, we have that:

1) v/v’ = L/L’ = sin(Pi/4) = cos(Pi/4) = 2^(1/2)/2.

2) p/p’ = W/W’ = tan(Pi/4) = cot(Pi/4) = 1.

3) E/E’ = w/w’ = sec(Pi/4) = csc(Pi/4) = 2^(1/2).

Otherwise if v < b we have that:

v/v’ < L/L’ and p/p’ < W/W’ and E/E’ < w/w’.

Otherwise if v > b we have that:

v/v’ > L/L’ and p/p’ > W/W’ and E/E’ > w/w’.

The corresponding imaginary part is in A232738.

This is the maximum increase in mass-energy a particle can carry away from a neutral rotating (Kerr) black hole via the Penrose process.

Apart from leading digits the same as A174968, A157214 and A010503. - R. J. Mathar, Feb 24 2016

Define P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k) for n >= 1, and P(0) = 1, with x(q) = C1 and x(n) = 1 for all other n. We find that C2 = lim_{n -> infinity} P(n) = exp((C1-1)/q).

The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.

Some transform pairs: C1 = A002162 (log(2)) and C2 = A135002 (2/exp(1)); C1 = A016627 (log(4)) and C2 = A135004 (4/exp(1)); C1 = A001113 (exp(1)) and C2 = A234473 (exp(exp(1)-1)).

From Peter Bala, Oct 23 2019: (Start)

The constant is irrational: Henry Cohn gives the following proof in Todd and Vishals Blog - "By the way, here's my favorite application of the tanh continued fraction: exp(sqrt(2)) is irrational.

Consider sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1). If exp(sqrt(2)) were rational, or even in Q(sqrt(2)), then this expression would be in Q(sqrt(2)). However, it is sqrt(2)*tanh(1/sqrt(2)), and the tanh continued fraction shows that this equals [0,1,6,5,14,9,22,13,...]. If it were in Q(sqrt(2)), it would have a periodic simple continued fraction expansion, but it doesn't." (End)

Every row intersperses all other rows, and every column intersperses all other columns. The array is the dispersion of the complement of (column 1 = A022776).

R(n,m) = position of n*r + m when all the numbers k*r + h, where r = sqrt(2), k >= 1, h >= 0, are jointly ranked. - Clark Kimberling, Oct 06 2017