A309767 Approximation of the 2-adic integer arctanh(4) up to 2^n.

0, 0, 0, 4, 4, 4, 4, 68, 196, 196, 708, 1732, 3780, 7876, 7876, 7876, 40644, 106180, 237252, 499396, 1023684, 1023684, 3120836, 7315140, 15703748, 15703748, 49258180, 116367044, 250584772, 250584772, 787455684, 787455684, 2934939332, 2934939332, 2934939332

Offset

0,4

Comments

arctanh(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-3)/4)} 4^(2*i+1)/(2*i+1)) mod 2^n.

Example

a(3) = 4^1 mod 2^3 = 4;
a(6) = 4^1 mod 2^6 = 4
a(7) = (4^1 + 4^3/3) mod 2^7 = 68;
a(10) = (4^1 + 4^3/3) mod 2^10 = 708;
a(11) = (4^1 + 4^3/3 + 4^5/5) mod 2^11 = 1732;
a(14) = (4^1 + 4^3/3 + 4^5/5) mod 2^14 = 7876;
a(15) = (4^1 + 4^3/3 + 4^5/5 + 4^7/7) mod 2^15 = 7876.
a(18) = (4^1 + 4^3/3 + 4^5/5 + 4^7/7) mod 2^18 = 237252.

Prog

(PARI) a(n) = lift(sum(i=0, (n-3)/4, Mod(4^(2*i+1)/(2*i+1), 2^n)))

Crossrefs

Cf. A309753, A309756, A309768.

Sequence in context: A309500 A321611 A309756 * A199500 A164838 A059112

Adjacent sequences: A309764 A309765 A309766 * A309768 A309769 A309770

Keyword

nonn

Author

Jianing Song, Aug 16 2019

Status

approved