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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). 7
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 (list; graph; refs; listen; history; text; internal format)
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

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.

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+=[b, ]

          a=2*b**2 + b + 4

          mod*=3

      return ary

print a271222(100) # Indranil Ghosh, Aug 04 2017, after Ruby

CROSSREFS

Cf. A268922, A268924, A271223, A271224.

Sequence in context: A154918 A176862 A176081 * A073339 A180092 A074636

Adjacent sequences:  A271219 A271220 A271221 * A271223 A271224 A271225

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Apr 05 2016

STATUS

approved

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Last modified January 29 14:07 EST 2020. Contains 331338 sequences. (Running on oeis4.)