A290559 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(2). These are the numbers congruent to 4 mod 7 (except for the initial 0).

0, 4, 39, 235, 235, 12240, 79468, 667713, 3961885, 15491487, 15491487, 15491487, 7924798459, 77131234464, 561576286499, 4630914723593, 23621160763365, 189785813611370, 1352938383547405, 4609765579368303, 4609765579368303, 403571097067428308

Offset: 0

Author

Seiichi Manyama, Aug 05 2017

Keywords

• nonn
• easy

Comments

x = ...450454,

x^2 = ...000002 = 2.

Examples

			a(1) = (    4)_7 = 4,
a(2) = (   54)_7 = 39,
a(3) = (  454)_7 = 235,
a(4) = (  454)_7 = 235,
a(5) = (50454)_7 = 12240.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1183
• Wikipedia, Hensel's Lemma.

Crossrefs

Cf. A051277, A290557, A290558.

Programs

PARI

a(n) = if (n==0, 0, 7^n - truncate(sqrt(2+O(7^n)))); \\ Michel Marcus, Aug 06 2017

Formula

If n > 0, a(n) = 7^n - A290557(n).

a(0) = 0 and a(1) = 4, a(n) = a(n-1) + 6 * (a(n-1)^2 - 2) mod 7^n for n > 1.

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