cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A320815 Digits of the 2-adic integer exp(4).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0
Offset: 0

Views

Author

Jianing Song, Oct 21 2018

Keywords

Comments

See A320814 for detailed information.
This is the multiplicative inverse of A321692. - Jianing Song, Nov 17 2018

Examples

			exp(4) = ...00111001000111000100011100100000101001101.

Links

Crossrefs

Cf. A320814, A321692.

Programs

  • PARI
    a(n) = lift(sum(i=0, n-(n>=2), Mod(4^i/i!, 2^(n+1))))\2^n
  • PARI
    a(n) = lift(exp(4 + O(2^(n+1))))\2^n; \\ Jianing Song, Nov 17 2018

Formula

a(n) = (A320814(n+1) - A320814(n))/2^n.

A321689 Approximation of the 2-adic integer exp(-4) up to 2^n.

Original entry on oeis.org

0, 1, 1, 5, 5, 5, 5, 5, 133, 389, 901, 1925, 3973, 8069, 8069, 24453, 57221, 57221, 188293, 450437, 974725, 974725, 974725, 974725, 974725, 17751941, 17751941, 84860805, 84860805, 84860805, 621731717, 621731717, 621731717, 4916699013, 4916699013
Offset: 0

Views

Author

Jianing Song, Nov 17 2018

Keywords

Comments

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

Examples

			A320840(1) = 1, (-4)^0/0! = 1, so a(1) = 1.
A320840(3) = 2, Sum_{i=0..1} (-4)^i/i! = -3 == 5 (mod 8), so a(3) = 5.
A320840(8) = 6, Sum_{i=0..5} (-4)^i/i! = -53/15 == 133 (mod 256), so a(8) = 133.
A320840(9) = 7, Sum_{i=0..6} (-4)^i/i! = 97/45 == 389 (mod 512), so a(9) = 389.
A320840(10) = 9, Sum_{i=0..8} (-4)^i/i! = 167/315 == 901 (mod 1024), so a(10) = 901.

Links

Crossrefs

Cf. A320814, A320840, A321692.

Programs

  • PARI
    a(n) = lift(sum(i=0, n-1-(n>=2), Mod((-4)^i/i!, 2^n)))
  • PARI
    a(n) = lift(exp(-4 + O(2^n)));

Formula

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.
a(n) = Sum_{i=0..n-1} A321692(i)*2^i.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.