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
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Apr 05 2016
STATUS
approved