A309756 Approximation of the 2-adic integer arctan(4) up to 2^n.

0, 0, 0, 4, 4, 4, 4, 68, 68, 324, 324, 324, 2372, 2372, 2372, 18756, 51524, 51524, 182596, 444740, 969028, 2017604, 4114756, 4114756, 4114756, 20891972, 20891972, 20891972, 20891972, 289327428, 289327428, 1363069252, 1363069252, 1363069252, 1363069252

Offset

0,4

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-3)/4)} (-1)^i*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 = 324;
a(11) = (4^1 - 4^3/3 + 4^5/5) mod 2^11 = 324;
a(14) = (4^1 - 4^3/3 + 4^5/5) mod 2^14 = 2372;
a(15) = (4^1 - 4^3/3 + 4^5/5 - 4^7/7) mod 2^15 = 18756.
a(18) = (4^1 - 4^3/3 + 4^5/5 - 4^7/7) mod 2^18 = 182596.

Prog

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

Crossrefs

Cf. A309751, A309766, A309767.

Sequence in context: A201981 A309500 A321611 * A309767 A199500 A164838

Adjacent sequences: A309753 A309754 A309755 * A309757 A309758 A309759

Keyword

nonn

Author

Jianing Song, Aug 16 2019

Status

approved