OFFSET
0,1
COMMENTS
See A048898 for the successive approximations to this 5-adic integer, called there u.
The digits of -u, the other 5-adic integer sqrt(-1), are given in A210851.
a(n) is the unique solution of the linear congruence 2*A048898(n)*a(n) + A210848(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)=2 follows from the formula given below.
a(n) + A210851(n) = 4 for n >= 1. - Robert Israel, Mar 04 2016
From Jianing Song, Sep 06 2022: (Start)
With a(0) = 1, 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 1 modulo 5.
With a(0) = 3, 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 3 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,2)}, where L(n,x) denotes the n-th Lucas polynomial, the n-th row polynomial of A114525. - Peter Bala, Dec 02 2022
LINKS
FORMULA
EXAMPLE
MAPLE
R:= select(t -> padic:-ratvaluep(t, 1)=2, [padic:-rootp(x^2+1, 5, 10001)]):
op([1, 1, 3], R); # Robert Israel, Mar 04 2016
MATHEMATICA
Table[Floor[First@Select[PowerModList[-1, 1/2, 5^(k+1)], Mod[#, 5]==2&]/5^k], {k, 0, 99}] (* Giorgos Kalogeropoulos, Feb 28 2023 *)
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