A392148 - OEIS

%I #20 Apr 24 2026 10:37:06

%S 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,

%T 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,

%U 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

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

%C 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, ...].

%C Related quasi-periodic continued fraction expansions:

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

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

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

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

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

%H Paolo Xausa, <a href="/A392148/b392148.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

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

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

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

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

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

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

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

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

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

%e 1.2874269452054219109029315844780560045853817799817751773978615048 ...

%p A392148 := tan(1/4) + sec(1/4):

%p s := convert(evalf(A392148, 140), string):

%p parse(s[1]), seq(parse(s[i+1]), i = 2..106);

%t First[RealDigits[Tan[1/4] + Sec[1/4], 10, 100]] (* _Paolo Xausa_, Feb 16 2026 *)

%o (PARI) tan(1/4)+1/cos(1/4) \\ _Charles R Greathouse IV_, Apr 24 2026

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

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

%K nonn,cons,easy

%O 1,2

%A _Peter Bala_, Jan 02 2026