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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).
10
0, 3, 10, 108, 2166, 4567, 38181, 155830, 1802916, 24862120, 266983762, 1961835256, 5916488742, 19757775943, 116646786350, 116646786350, 9611769806236, 42844700375837, 275475214363044, 6789129606004840, 75182500718243698, 154974767015855699
OFFSET
0,2
COMMENTS
x = ...216213,
x^2 = ...000002 = 2.
LINKS
Wikipedia, Hensel's Lemma.
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
EXAMPLE
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.
PROG
(PARI) a(n) = truncate(sqrt(2+O(7^(n)))); \\ Michel Marcus, Aug 06 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Aug 05 2017
STATUS
approved