The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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