A309751 Approximation of the 2-adic integer arctan(2) up to 2^n.

0, 0, 2, 2, 10, 10, 10, 74, 202, 202, 714, 714, 714, 714, 8906, 25290, 58058, 123594, 254666, 516810, 516810, 1565386, 1565386, 5759690, 14148298, 14148298, 47702730, 47702730, 181920458, 450355914, 987226826, 987226826, 3134710474, 7429677770, 7429677770

Offset

0,3

Comments

arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...

Links

Table of n, a(n) for n=0..34.

Wikipedia, p-adic number

Formula

a(n) = (Sum_{i=0..floor(n/2)-1} (-1)^i*2^(2*i+1)/(2*i+1)) mod 2^n.

Example

a(2) = 2^1 mod 2^2 = 2;
a(3) = 2^1 mod 2^3 = 2;
a(4) = (2^1 - 2^3/3) mod 2^4 = 2;
a(5) = (2^1 - 2^3/3) mod 2^5 = 10;
a(6) = (2^1 - 2^3/3 + 2^5/5) mod 2^6 = 10;
a(7) = (2^1 - 2^3/3 + 2^5/5) mod 2^7 = 74.

Prog

(PARI) a(n) = lift(sum(i=0, n/2-1, Mod((-1)^i*2^(2*i+1)/(2*i+1), 2^n)))

Crossrefs

Cf. A309752, A309753.

Sequence in context: A141610 A019241 A168295 * A249152 A216708 A032005

Adjacent sequences: A309748 A309749 A309750 * A309752 A309753 A309754

Keyword

nonn

Author

Jianing Song, Aug 15 2019

Extensions

Offset corrected by Georg Fischer, Jun 22 2022

Status

approved