A290558 Coefficients in 7-adic expansion of sqrt(2).

4, 5, 4, 0, 5, 4, 5, 4, 2, 0, 0, 4, 5, 5, 6, 4, 5, 5, 2, 0, 5, 3, 4, 0, 0, 3, 1, 1, 0, 3, 2, 1, 6, 5, 0, 3, 6, 2, 0, 4, 2, 2, 0, 2, 4, 2, 2, 4, 0, 5, 3, 2, 5, 3, 5, 2, 4, 0, 0, 6, 3, 1, 1, 5, 5, 4, 6, 0, 0, 5, 5, 4, 2, 2, 2, 4, 3, 0, 0, 3, 0, 5, 2, 2, 4, 4, 5, 3

Offset: 0

Author

Seiichi Manyama, Aug 05 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Peter Bala, Using Chebyshev polynomials to find the p-adic square roots of 2 and 3, Dec 2022.

Crossrefs

Cf. A051277, A290559.

Programs

Maple

t := proc(n) option remember; if n = 1 then 4 else irem(t(n-1)^7 - 7*t(n-1)^5 + 14*t(n-1)^3 - 7*t(n-1), 7^n) end if; end:
convert(t(100), base, 7); # Peter Bala, Nov 20 2022

PARI

{ my(v=Vecrev( digits( truncate( (2+O(7^100))^(1/2) ), 7) )); vector(#v,k,6-v[k]+(k==1)) } \\ Joerg Arndt, Aug 06 2017

Ruby

require 'OpenSSL'
def f_a(ary, a)
  (0..ary.size - 1).inject(0){|s, i| s + ary[i] * a ** i}
end
def df(ary)
  (1..ary.size - 1).map{|i| i * ary[i]}
end
def A(c_ary, k, m, n)
  x = OpenSSL::BN.new((-f_a(df(c_ary), k)).to_s).mod_inverse(m).to_i % m
  f_ary = c_ary.map{|i| x * i}
  f_ary[1] += 1
  d_ary = []
  ary = [0]
  a, mod = k, m
  (n + 1).times{|i|
    b = a % mod
    d_ary << (b - ary[-1]) / m ** i
    ary << b
    a = f_a(f_ary, b)
    mod *= m
  }
  d_ary
end
def A290558(n)
  A([-2, 0, 1], 4, 7, n)
end
p A290558(100)

Formula

a(n) = 6 - A051277(n) for n > 0.

Equals the 7-adic limit as n -> oo of 2*T(7^n,2) = the 7-adic limit as n -> oo of (2 + sqrt(3))^(7^n) + (2 - sqrt(3))^(7^n), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Nov 20 2022