A143623 - OEIS

A143623

Decimal expansion of the constant cos(1) + sin(1).

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

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OFFSET

1,2

COMMENTS

cos(1) + sin(1) = Sum_{n >= 0} (-1)^floor(n/2)/n! = 1 + 1/1! - 1/2! - 1/3! + 1/4! + 1/5! - 1/6! - 1/7! + + - - ... .

Define E_2(k) = Sum_{n >= 0} (-1)^floor(n/2)*n^k/n! for k = 0, 1, 2, ... . Then E_2(0) = cos(1) + sin(1) and E_2(1) = cos(1) - sin(1).

Furthermore, E_2(k) is an integral linear combination of E_2(0) and E_2(1) (a Dobinski-type relation). For example, E_2(2) = E_2(1) - E_2(0), E_2(3) = -3*E_2(0) and E_2(4) = -5*E_2(1) - 6*E_2(0). The precise result is E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1).

The decimal expansion of the constant cos(1) - sin(1) = E_2(1) is recorded in A143624. Compare with A143625.

Transcendental by the Hermite-Lindemann theorem: expand into exponential form, note e^i is transcendental, note the two have an algebraic relationship. - Charles R Greathouse IV, Apr 10 2026

LINKS

Table of n, a(n) for n=1..101.

Index entries for transcendental numbers.

FORMULA

Equals sin(1+Pi/4)*sqrt(2). - Franklin T. Adams-Watters, Jun 27 2014

EXAMPLE

1.38177329067603622405 ... .

MATHEMATICA

RealDigits[Cos[1]+Sin[1], 10, 120][[1]] (* Harvey P. Dale, Mar 01 2019 *)

PROG

(PARI) sqrt(2)*sin(1+Pi/4) \\ Charles R Greathouse IV, Feb 03 2025

(PARI) sin(1)+cos(1) \\ Charles R Greathouse IV, Apr 10 2026