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A210851
Digits of one of the two 5-adic integers sqrt(-1).
26
3, 3, 2, 3, 1, 0, 2, 1, 4, 1, 2, 2, 4, 0, 3, 1, 2, 0, 4, 0, 1, 0, 4, 0, 3, 2, 0, 3, 0, 3, 3, 1, 3, 0, 3, 0, 2, 4, 3, 3, 1, 1, 2, 2, 0, 4, 0, 2, 0, 4, 1, 3, 2, 0, 4, 1, 1, 4, 1, 4, 4, 4, 1, 3, 1, 3, 3, 4, 1, 4, 4, 1, 0, 3, 1, 1, 1, 0, 4, 2, 2, 4, 2, 4, 3, 4, 0, 3, 3, 0, 0, 2, 3, 4, 2, 4, 4, 1, 4, 0
OFFSET
0,1
COMMENTS
See A048899 for the successive approximations to this 5-adic integer, called -u in a comment on A048898.
The digits of u, the other 5-adic integer sqrt(-1), are given in A210850.
a(n) is the (unique) solution of the linear congruence 2*A048899(n)*a(n) + A210849(n) == 0 (mod 5), n >= 1. Therefore only the values 0, 1, 2, 3 and 4 appear. See the Nagell reference given in A210848, eq. (6) on p. 86 adapted to this case. a(0)=3 follows from the formula given below.
If n>0, a(n) == -(A210849(n)) (mod 5), since A048899(n) == 3 (mod 5). - Álvar Ibeas, Feb 21 2017
If a(n)=0 then A048899(n+1) and A048899(n) coincide.
From Jianing Song, Sep 06 2022: (Start)
With a(0) = 2, this is the digits of one of the four 4th root of -4 in the ring of 5-adic integers, the one that is congruent to 2 modulo 5.
With a(0) = 4, this is the digits of one of the four 4th root of -4 in the ring of 5-adic integers, the one that is congruent to 4 modulo 5. (End)
This square root of -1 in the 5-adic integers is equal to the 5-adic limit of the sequence {L(5^n,3)}, where L(n,x) denotes the n-th Lucas polynomial, the n-th row polynomial of A114525. - Peter Bala, Dec 02 2022
FORMULA
a(n) = (b(n+1) - b(n))/5^n, n >= 0, with b(n):=A048899(n) computed from its recurrence. A Maple program for b(n) is given there.
A048899(n+1) = Sum_{k=0..n} a(k)*5^k, n >= 0.
EXAMPLE
a(3) = 3 because 2*68*3 + 37 == 0 (mod 5).
A048899(4) = 443 = 3*5^0 + 3*5^1 + 2*5^2 + 3*5^3.
a(5) = 0 because A048899(6) = A048899(5) = 3*5^0 + 3*5^1 + 2*5^2 + 3*5^3 + 1*5^4 = 1068.
MAPLE
R:= select(t -> padic:-ratvaluep(t, 1)=3, [padic:-rootp(x^2+1, 5, 200)]):
op([1, 1, 3], R); # Robert Israel, Mar 04 2016
PROG
(PARI) a(n) = truncate(-sqrt(-1+O(5^(n+1))))\5^n; \\ Michel Marcus, Mar 05 2016
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Wolfdieter Lang, Apr 30 2012
EXTENSIONS
Keyword "base" added by Jianing Song, Feb 17 2021
STATUS
approved