OFFSET
0,2
COMMENTS
The other approximation for the 5-adic integer sqrt(-4) with numbers 1 (mod 5) is given in A268922. See this also for more details and references.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1432
FORMULA
Recurrence for n >= 1: a(n) = modp( a(n-1) + 3*(a(n-1)^2 + 4), 5^n), n >= 2, with a(1) = 4. 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) = 5^n - A268922(n), n >= 1.
a(n) == Lucas(3*(5^n)) (mod 5^n). - Peter Bala, Nov 14 2022
MAPLE
with(padic): D2:=op(3, op([evalp(RootOf(x^2+4), 5, 20)][2])):
0, seq(sum('D2[k]*5^(k-1)', 'k'=1..n), n=1..20);
# alternative program
a := proc(n) option remember; if n = 1 then 4 else irem( a(n-1)^5 + 5*a(n-1)^3 + 5*a(n-1), 5^n) end if; end: seq(a(n), n = 1..20); # Peter Bala, Nov 14 2022
PROG
(PARI) a(n) = if (n==0, 0, 5^n - truncate(sqrt(-4+O(5^(n))))); \\ Michel Marcus, Mar 07 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Mar 02 2016
STATUS
approved