A091661 Coefficients in a 10-adic square root of 1.

9, 4, 2, 1, 8, 7, 5, 2, 4, 6, 3, 8, 9, 1, 5, 2, 1, 5, 4, 8, 7, 4, 5, 9, 9, 3, 2, 3, 1, 2, 8, 0, 0, 8, 1, 2, 2, 9, 7, 1, 6, 4, 6, 4, 8, 6, 4, 8, 4, 1, 1, 1, 0, 0, 2, 2, 6, 7, 2, 7, 1, 6, 1, 9, 1, 0, 3, 3, 3, 4, 2, 1, 0, 8, 7, 9, 1, 0, 7, 7, 8, 5, 0, 6, 9, 3, 3, 6, 1, 2, 8, 3, 6, 4, 1, 0, 6, 0, 9, 7

Offset: 0

Author

Edoardo Gueglio (egueglio(AT)yahoo.it), Jan 28 2004

Keywords

• base
• nonn

Comments

10-adic integer x=.....239954784512519836425781249 satisfying x^3 = x.

Let a,b be integers defined in A018247, A018248 satisfying a^2=a, b^2=b, obviously a^3=a, b^3=b; let c,d,e,f be integers defined in A091661, A063006, A091663, A091664 then c^3=c, d^3=d, e^3=e, f^3=f, c+d=1, a+e=1, b+f=1, b+c=a, d+f=e, a+f=c, a=f+1, b=e+1, cd=-1, af=-1, gh=-1 where -1=.....999999999.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..9999

Crossrefs

Another 10-adic root of 1 is given by A063006.

Cf. A018247, A018248.

Programs

Mathematica

To calculate c, d, e, f use Mathematica algorithms for a, b and equations: c=a-b, d=1-c, e=b-1, f=a-1.

Ruby

def A(s, n)
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k = j * m + s
      if (k ** 2 - k) % (m * 10) == 0
        s = k
        break
      end
    }
  }
  s
end
def A091661(n)
  str = (10 ** (n + 1) + A(5, n) - A(6, n)).to_s.reverse
  (0..n).map{|i| str[i].to_i}
end
p A091661(100) # Seiichi Manyama, Jul 31 2017

Formula

For n>0, a(n) = 9 - A063006(n).