A392148 Decimal expansion of tan(1/4) + sec(1/4).

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

Offset

1,2

Comments

The simple continued fraction expansion of tan(1/4) + sec(1/4) equals [1; 3, 2, 11, 2, 19, 2, 27, 2, ..., 8*k + 3, 2, ...].

Related quasi-periodic continued fraction expansions:

sec(1/4) - tan(1/4) = [0; 1, 3, 2, 11, 2, 19, 2, 27, 2, ..., 8*k + 3, 2, ...].

4*(tan(1/4) + sec(1/4)) = [5; {6, 1, 4*k+2, 8, 4*k+4, 1}, ...], k >= 0.

2*(tan(1/4) + sec(1/4)) = [2; 1, {1, 2, 1, 8*k+5, 4, 8*k+9}, ...], k >=0

(1/2)*(tan(1/4) + sec(1/4)) = [0; 1, 1, {8*k+1, 4, 8*k+5, 1, 2, 1}, ...], k >= 0

(1/4)*(tan(1/4) + sec(1/4)) = [0; 3, 9, {4*k+2, 1, 6, 1, 4*k+4, 8}, ...], k >= 0.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Index entries for transcendental numbers.

Formula

Constant c = tan(1/8 + Pi/4).

c = exp(arctanh(sin(1/4))).

c = -i*(exp(i/4) + i)/(exp(i/4) - i), where i = sqrt(-1).

c = (cos(1/8) + sin(1/8))/(cos(1/8) - sin(1/8)) = sqrt((1 + sin(1/4))/(1 - sin(1/4))).

(c - 1)/(c + 1) = tan(1/8).

c - 1/c = 2*tan(1/4).

c + 1/c = 2*sec(1/4).

c = exp( Integral_{x = 0..1/4} sec(x) dx).

c = 5/4 + Integral_{x = 0..1/4} sin(x)/(1 - sin(x)) dx.

Example

1.2874269452054219109029315844780560045853817799817751773978615048 ...

Maple

A392148 := tan(1/4) + sec(1/4):
s := convert(evalf(A392148, 140), string):
parse(s[1]), seq(parse(s[i+1]), i = 2..106);

Mathematica

First[RealDigits[Tan[1/4] + Sec[1/4], 10, 100]] (* Paolo Xausa, Feb 16 2026 *)

Prog

(PARI) tan(1/4)+1/cos(1/4) \\ Charles R Greathouse IV, Apr 24 2026

Crossrefs

(tan(1/k) + sec(1/k)): A392146 (k = 2), A392147 (k = 3), A392149 (k = 5), A392150 (k = 6).

Cf. A019427, A133265, A161013, A384932, A392151, A392152, A392153.

Sequence in context: A021352 A094289 A051429 * A268308 A027606 A202693

Adjacent sequences: A392145 A392146 A392147 * A392149 A392150 A392151

Keyword

nonn,cons,easy

Author

Peter Bala, Jan 02 2026

Status

approved