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

%I #9 Nov 18 2018 10:03:52

%S 0,1,1,5,5,5,5,5,133,389,901,1925,3973,8069,8069,24453,57221,57221,

%T 188293,450437,974725,974725,974725,974725,974725,17751941,17751941,

%U 84860805,84860805,84860805,621731717,621731717,621731717,4916699013,4916699013

%N Approximation of the 2-adic integer exp(-4) up to 2^n.

%C Let 4Q_2 = {x belongs to Q_2 : |x|_2 <= 1/4} and 4Q_2 + 1 = {x belongs to Q_2: |x - 1|_2 <= 1/4}. Define exp(x) = Sum_{k>=0} x^k/k! and log(x) = -Sum_{k>=0} (1 - x)^k/k over 2-adic field, then exp(x) is a one-to-one mapping from 4Q_2 to 4Q_2 + 1, and log(x) is the inverse of exp(x).

%C a(n) is the multiplicative inverse of A320814(n) modulo 2^n.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">p-adic number</a>

%F If Sum_{i=0..A320840(n)-1} (-4)^i/i! = p/q, gcd(p, q) = 1, then a(n) = p*q^(-1) mod 2^n.

%F a(n) = Sum_{i=0..n-1} A321692(i)*2^i.

%e A320840(1) = 1, (-4)^0/0! = 1, so a(1) = 1.

%e A320840(3) = 2, Sum_{i=0..1} (-4)^i/i! = -3 == 5 (mod 8), so a(3) = 5.

%e A320840(8) = 6, Sum_{i=0..5} (-4)^i/i! = -53/15 == 133 (mod 256), so a(8) = 133.

%e A320840(9) = 7, Sum_{i=0..6} (-4)^i/i! = 97/45 == 389 (mod 512), so a(9) = 389.

%e A320840(10) = 9, Sum_{i=0..8} (-4)^i/i! = 167/315 == 901 (mod 1024), so a(10) = 901.

%o (PARI) a(n) = lift(sum(i=0, n-1-(n>=2), Mod((-4)^i/i!, 2^n)))

%o (PARI) a(n) = lift(exp(-4 + O(2^n)));

%Y Cf. A320814, A320840, A321692.

%K nonn

%O 0,4

%A _Jianing Song_, Nov 17 2018