A271222 One of the two successive approximations up to 3^n for the 3-adic integer sqrt(-2). These are the numbers congruent to 2 mod 3 (except for the initial 0).

0, 2, 5, 5, 59, 221, 221, 1679, 3866, 16988, 56354, 174452, 174452, 705893, 705893, 10271831, 24620738, 110714180, 239854343, 627274832, 2951797766, 2951797766, 2951797766, 65713916984, 159857095811, 442286632292

Offset

0,2

Comments

The other approximation for the 3-adic integer sqrt(-2) with numbers 1 (mod 3) is given in A268924.

For the digits of this 3-adic integer sqrt(-2), that is the scaled first differences, see A271224. This 3-adic number has the digits read from the right to the left ... 20020121022200011120021121201022212022012 = -u. For the digits of u see A271223.

For details see the W. Lang link ``Note on a Recurrence or Approximation Sequences of p-adic Square Roots'' given under A268922, also for the Nagell reference and Hensel lifting. Here p = 3, b = 2, x_2 = 2 and z_2 = 2.

References

Trygve Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, p. 87.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..2095

Peter Bala, A note on A268924 and A271222, Nov 28 2022.

Wikipedia, Hensel's Lemma.

Formula

a(n)^2 + 2 == 0 (mod 3^n), and a(n) == 2 (mod 3). Representatives of the complete residue system {0, 1, ..., 3^n-1} are taken.

Recurrence for n >= 1: a(n) = modp(a(n-1) + 2*(a(n-1)^2 + 2), 3^n), n >= 2, with a(1) = 2. Here modp(a, m) is used to pick the representative of the residue class a modulo m from the smallest nonnegative complete residue system {0, 1, ... , m-1}.

a(n) = 3^n - A268924(n), n >= 1.

a(n) == A002203(3^n) (mod 3^n). - Peter Bala, Nov 10 2022

Example

n=2: 5^2 + 2 = 27 == 0 (mod 3^2), and 5 is the only solution from {0, 1, ..., 8} which is congruent to 2 modulo 3.
n=3: the only solution of X^2 + 2 == 0 (mod 3^3) with X from {0, ..., 26} and X == 2(mod 3) is 5. The number 22 = A268924(3) also satisfies the first congruence but not the second one: 22  == 1 (mod 3).
n=4: the only solution of X^2 + 2 == 0 (mod 3^4) with X from {0, ..., 80} and X == 2 (mod 3) is 59. The number 22 = A268924(4) also satisfies the first congruence but not the second one: 59  == 1 (mod 3).

Maple

with(padic):D2:=op(3, op([evalp(RootOf(x^2+2), 3, 20)][2])): 0, seq(sum('D2[k]*3^(k-1)', 'k'=1..n), n=1..20);

Prog

(PARI) a(n) = if (n, 3^n - truncate(sqrt(-2+O(3^(n)))), 0); \\ Michel Marcus, Apr 09 2016
(Ruby)
def A271222(n)
  ary = [0]
  a, mod = 2, 3
  n.times{
    b = a % mod
    ary << b
    a = 2 * b * b + b + 4
    mod *= 3
  }
  ary
end
p A271222(100) # Seiichi Manyama, Aug 03 2017
(Python)
def a271222(n):
      ary=[0]
      a, mod = 2, 3
      for i in range(n):
          b=a%mod
          ary.append(b)
          a=2*b**2 + b + 4
          mod*=3
      return ary
print(a271222(100)) # Indranil Ghosh, Aug 04 2017, after Ruby

Crossrefs

Cf. A002293, A268922, A268924, A271223, A271224.

Sequence in context: A176862 A352392 A176081 * A073339 A180092 A074636

Adjacent sequences: A271219 A271220 A271221 * A271223 A271224 A271225

Keyword

nonn,easy

Author

Wolfdieter Lang, Apr 05 2016

Status

approved