A309767 - OEIS

%I #7 Aug 18 2019 04:20:50

%S 0,0,0,4,4,4,4,68,196,196,708,1732,3780,7876,7876,7876,40644,106180,

%T 237252,499396,1023684,1023684,3120836,7315140,15703748,15703748,

%U 49258180,116367044,250584772,250584772,787455684,787455684,2934939332,2934939332,2934939332

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

%C arctanh(x) = x + x^3/3 + x^5/5 + x^7/7 + ...

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

%F a(n) = (Sum_{i=0..floor((n-3)/4)} 4^(2*i+1)/(2*i+1)) mod 2^n.

%e a(3) = 4^1 mod 2^3 = 4;

%e a(6) = 4^1 mod 2^6 = 4

%e a(7) = (4^1 + 4^3/3) mod 2^7 = 68;

%e a(10) = (4^1 + 4^3/3) mod 2^10 = 708;

%e a(11) = (4^1 + 4^3/3 + 4^5/5) mod 2^11 = 1732;

%e a(14) = (4^1 + 4^3/3 + 4^5/5) mod 2^14 = 7876;

%e a(15) = (4^1 + 4^3/3 + 4^5/5 + 4^7/7) mod 2^15 = 7876.

%e a(18) = (4^1 + 4^3/3 + 4^5/5 + 4^7/7) mod 2^18 = 237252.

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

%Y Cf. A309753, A309756, A309768.

%K nonn

%O 0,4

%A _Jianing Song_, Aug 16 2019