A392151 Decimal expansion of tan(2) - sec(2).

2, 1, 7, 9, 5, 8, 0, 9, 8, 4, 6, 0, 8, 6, 1, 9, 9, 8, 1, 1, 1, 2, 9, 4, 2, 9, 9, 1, 0, 6, 3, 8, 3, 6, 8, 2, 8, 1, 3, 1, 0, 3, 3, 4, 6, 9, 2, 6, 8, 6, 2, 8, 4, 8, 4, 8, 1, 4, 4, 5, 1, 0, 0, 2, 4, 5, 3, 0, 0, 3, 3, 3, 8, 3, 6, 3, 0, 4, 2, 0, 4, 6, 0, 4, 0, 6, 6, 5, 4, 5, 8, 9, 3, 5, 2, 7, 6, 3, 1, 9, 5, 5, 3, 1, 5

Offset

0,1

Comments

The simple continued fraction expansion of tan(2) - sec(2) = [0; 4, {1, 3*k+1, 2, 3*k+2, 1, 12*k+16}], where k >= 0 in the quasi-period of length 6.

Related simple continued fraction expansions:

-(tan(2) + sec(2)) = [4; {3*k+1, 2, 3*k+2, 1, 12*k+16}]. See A100261.

2*(tan(2) - sec(2)) = [0; 2, 3, 2, 2, {48*k+34, 1, 3*k+1, 1, 2, 1, 3*k+2, 2, 12*k+14, 2, 3*(k+1), 1, 2, 1, 3*k+3, 1}], where k >= 0 in the quasi-period of length 16.

(1/2)*(tan(2) - sec(2)) = [0; 9, 5, 1, {2, 12*k+8, 2, 3*k+2, 4, 3*k+2, 1, 48*k+58, 1, 3*k+3, 4, 3*k+4}], where k >= 0 in the quasi-period of length 12.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

A. J. van der Poorten, Continued fraction expansions of values of the exponential function and related fun with continued fractions, Nieuw Archief voor Wiskunde, Vol. 14 (1996), pp. 221-230 (see Theorem 2).

Index entries for transcendental numbers.

Formula

c: = tan(2) - sec(2) = tan(1 - Pi/4) = -cot(1 + Pi/4) = (tan(1) - 1)/(tan(1) + 1).

c = sqrt((1 - sin(2))/(1 + sin(2))) = (sin(1) - cos(1))/(sin(1) + cos(1)) = -A143624/A143623.

c = -i * (exp(2*i) - i)/(exp(2*i) + i), where i = sqrt(-1).

c = -exp(-2*arctanh(tan(1))).

(1 + c)/(1 - c) = tan(1) = A049471.

c - 1/c = 2*tan(2).

c + 1/c = -2*sec(2).

c = - exp( - Integral_{x = 0..2} sec(x) dx ).

Example

0.2179580984608619981112942991063836828131033469268628484814451002453003338363042046...

Maple

A392151 := tan(2) - sec(2):
s := convert(evalf(A392151, 110), string):
seq( parse(s[i]), i = 2..106);

Mathematica

First[RealDigits[Tan[2] - Sec[2], 10, 100]] (* Paolo Xausa, Feb 16 2026 *)

Prog

(PARI) tan(2)-1/cos(2) \\ Charles R Greathouse IV, Apr 24 2026

Crossrefs

Cf. A049471, A100261, A133265, A143623, A143624, A384932, A392146, A392147, A392148, A392149, A392150.

Sequence in context: A141513 A258058 A238223 * A102817 A026252 A032298

Adjacent sequences: A392148 A392149 A392150 * A392152 A392153 A392154

Keyword

nonn,cons,easy

Author

Peter Bala, Jan 04 2026

Status

approved