A290557 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(2). These are the numbers congruent to 3 mod 7 (except for the initial 0).

0, 3, 10, 108, 2166, 4567, 38181, 155830, 1802916, 24862120, 266983762, 1961835256, 5916488742, 19757775943, 116646786350, 116646786350, 9611769806236, 42844700375837, 275475214363044, 6789129606004840, 75182500718243698, 154974767015855699

Offset: 0

Author

Seiichi Manyama, Aug 05 2017

Keywords

• nonn
• easy

Comments

x = ...216213,

x^2 = ...000002 = 2.

Examples

			a(1) = (    3)_7 = 3,
a(2) = (   13)_7 = 10,
a(3) = (  213)_7 = 108,
a(4) = ( 6213)_7 = 2166,
a(5) = (16213)_7 = 4567.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1184
• Wikipedia, Hensel's Lemma.

Crossrefs

Cf. A034945, A051277, A290559.

Programs

PARI

a(n) = truncate(sqrt(2+O(7^(n)))); \\ Michel Marcus, Aug 06 2017

PARI

a(n) = lift(sqrt(2 + O(7^n))); \\ Robert L. Brown, Jun 16 2025

Formula

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

a(n) == 2*T(7^n, 3/2) (mod 7^n) == ((3 + sqrt(5))/2)^(7^n) + ((3 - sqrt(5))/2)^(7^n) (mod 7^n), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Dec 03 2022