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Plouffe's Constants

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Plouffe's constants are numbers arising in summations of series related to r_n=f(2^n) where f is a trigonometric function and n is a nonnegative integer. Define the indicator function using the Iverson bracket

rho(x)={1 for x<0; 0 for x>=0.
(1)
PlouffesConstantA

Now define a_n through

a_n=sin(2^n)
(2)
={sin1 for n=0; 2a_0sqrt(1-a_0^2) for n=1; 2a_(n-1)(1-2a_(n-2)^2) for n>=2,
(3)

then

A=sum_(n=0)^(infty)(rho(a_n))/(2^(n+1))
(4)
=1/(2pi)
(5)
=0.15915...
(6)

(OEIS A086201).

PlouffesConstantB

For

b_n=cos(2^n)
(7)
={cos1 for n=0; 2b_(n-1)^2-1 for n>=1,
(8)

the sum is (amazingly) given by

B=sum_(n=0)^(infty)(rho(b_n))/(2^(n+1))
(9)
=1/pi direct sum 1/(2pi)
(10)
=0.4756260767...
(11)

(OEIS A086202), where direct sum denotes the XOR of binary digits (Chowdhury 2001a; Finch 2003, p. 432). A related sum, with b_(-1)=cos(1/2), is given by

B^'=sum_(n=0)^(infty)(rho(b_nb_(n-1)))/(2^(n+1))
(12)
=1/pi direct sum 1/(4pi)
(13)
=0.27007972...
(14)

(OEIS A111953), where direct sum again denotes the XOR of binary digits (Chowdhury 2001b; Finch 2024, §6.5).

PlouffesConstantC

Letting

c_n=tan(2^n)
(15)
={tan1 for n=0; (2c_(n-1))/(1-c_(n-1)^2) for n>=1,
(16)

then

C=sum_(n=0)^(infty)(rho(c_n))/(2^(n+1))
(17)
=1/pi
(18)

(OEIS A049541).

Plouffe asked if the above processes could be "inverted." He considered

alpha_n=sin(2^nsin^(-1)1/2)
(19)
={1/2 for n=0; 1/2sqrt(3) for n=1; 2alpha_(n-1)(1-2alpha_(n-2)^2) for n>=2,
(20)

giving

A=sum_(n=0)^(infty)(rho(alpha_n))/(2^(n+1))
(21)
=1/(12),
(22)

and

beta_n=cos(2^ncos^(-1)1/2)
(23)
={1/2 for n=0; 2beta_(n-1)^2-1 for n>=1,
(24)

giving

B=sum_(n=0)^(infty)(rho(beta_n))/(2^(n+1))
(25)
=1/2,
(26)

and

gamma_n=tan(2^ntan^(-1)1/2)
(27)
={1/2 for n=0; (2gamma_(n-1))/(1-gamma_(n-1)^2) for n>=1,
(28)

giving

C=sum_(n=0)^(infty)(rho(gamma_n))/(2^(n+1))
(29)
=1/pitan^(-1)(1/2)
(30)
=0.1475836...
(31)

(OEIS A086203), where the identity was conjectured by Plouffe and proved by Borwein and Girgensohn (1995).

C is sometimes known as Plouffe's constant (Plouffe 1998), although this angle had arisen in the geometry of the icosahedron dating back at least to the Ancient Greeks (Smith 2003).

Plouffe's constant is transcendental, as are all numbers of the form (tan^(-1)x)/pi for x rational and x!=0,+/-1 (Smith 2003, Margolius).

The positions of the 1s in the binary expansion of this constant are 3, 6, 8, 9, 10, 13, 21, 23, ... (OEIS A004715).

Note that the essential idea behind such binary expansions was already known as the "CORDIC" algorithms for computing inverse trigonometric functions (Volder 1959), which Smith (2003) suggests were anticipated by Archimedes, and has been the subject of numerous papers (Walther 1971) and implemented inside numerous commercial electronic calculators such as the HP-35 (Smith 2003).

Borwein and Girgensohn (1995) extended Plouffe's gamma_n to arbitrary real x, showing that if tangent poles are assigned the value -infty and

xi_n=tan(2^ntan^(-1)x)={x for n=0; 0 for n>=1 and xi_(n-1)=-infty; (2xi_(n-1))/(1-xi_(n-1)^2) for n>=1, xi_(n-1)!=-infty, and |xi_(n-1)|!=1; -infty for n>=1 and |xi_(n-1)|=1,
(32)

then

sum_(n=0)^infty(rho(xi_n))/(2^(n+1))={(tan^(-1)x)/pi for x>=0; 1+(tan^(-1)x)/pi for x<0.
(33)

Borwein and Girgensohn (1995) also give much more general recurrences and formulas.

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References

Borwein, J. M. and Girgensohn, R. "Addition Theorems and Binary Expansions." Canad. J. Math. 47, 262-273, 1995. https://doi.org/10.4153/CJM-1995-013-4.Chowdhury, M. "A Formula for 0.4756260767...." Unpublished note, 2001a.Chowdhury, M. "On Iterates of the Chaotic Logistic Function t_(n+1)=4t_n(1-t_n)." Unpublished note, 2001b.Finch, S. R. "Plouffe's Constant." §6.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 430-433, 2003.Finch, S. R. "Errata and Addenda to Mathematical Constants." 27 May 2024. https://arxiv.org/abs/2001.00578.Margolius, B. H. "Plouffe's Constant Is Transcendental." http://www.lacim.uqam.ca/~plouffe/articles/plouffe.pdf.Plouffe, S. "The Computation of Certain Numbers Using a Ruler and Compass." J. Integer Sequences 1, No. 98.1.3, 1998. https://cs.uwaterloo.ca/journals/JIS/compass.html.Sloane, N. J. A. Sequences A004715, A049541, A086201, A086202, A086203, and A111953 in "The On-Line Encyclopedia of Integer Sequences."Smith, W. D. "Pythagorean Triples, Rational Angles, and Space-Filling Simplices." 2003. https://plouffe.fr/simon/articles/Citations/Pythagorean%20triples%2C%20rational%20angles.pdf.Volder, J. "The CORDIC Trigonometric Computing Technique." IRE Trans. Elec. Comput. EC-8, 330-334, 1959.Walther, J. S. "A Unified Algorithm for Elementary Functions." In Spring Joint Computer Conference. pp. 379-385, 1971.

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Plouffe's Constants

Cite this as:

Weisstein, Eric W. "Plouffe's Constants." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PlouffesConstants.html

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