A290806 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-5). These are the numbers congruent to 3 mod 7 (except for the initial 0).

0, 3, 17, 311, 997, 3398, 20205, 608450, 2255536, 25314740, 25314740, 307789989, 8217096961, 77423532966, 368090564187, 4437429001281, 4437429001281, 4437429001281, 4437429001281, 3261264624822179, 3261264624822179, 3261264624822179, 1120352992791390193

Offset

0,2

Comments

x = ...112623,

x^2 = ...666662 = -5.

Links

Robert Israel, Table of n, a(n) for n = 0..1182

Wikipedia, Hensel's Lemma.

Formula

a(0) = 0 and a(1) = 3, a(n) = a(n-1) + (a(n-1)^2 + 5) mod 7^n for n > 1.

Example

a(1) =    3_7 = 3,
a(2) =   23_7 = 17,
a(3) =  623_7 = 311,
a(4) = 2623_7 = 997.

Maple

with(padic):
R:= [rootp(x^2+5, 7, 100)]:
R1:= op(select(t -> ratvaluep(evalp(t, 7, 1))=3, R)):
seq(ratvaluep(evalp(R1, 7, n)), n=0..100); # Robert Israel, Aug 13 2017

Prog

(PARI) a(n) = if (n, truncate(sqrt(-5+O(7^(n)))), 0)

Crossrefs

Cf. A290798, A290809.

Sequence in context: A155201 A062622 A271609 * A009592 A051294 A192556

Adjacent sequences: A290803 A290804 A290805 * A290807 A290808 A290809

Keyword

nonn

Author

Seiichi Manyama, Aug 11 2017

Status

approved