This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A290557 #32 Jun 16 2025 11:18:28 %S A290557 0,3,10,108,2166,4567,38181,155830,1802916,24862120,266983762, %T A290557 1961835256,5916488742,19757775943,116646786350,116646786350, %U A290557 9611769806236,42844700375837,275475214363044,6789129606004840,75182500718243698,154974767015855699 %N 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). %C A290557 x = ...216213, %C A290557 x^2 = ...000002 = 2. %H A290557 Seiichi Manyama, <a href="/A290557/b290557.txt">Table of n, a(n) for n = 0..1184</a> %H A290557 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>. %F A290557 a(0) = 0 and a(1) = 3, a(n) = a(n-1) + (a(n-1)^2 - 2) mod 7^n for n > 1. %F A290557 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 %e A290557 a(1) = ( 3)_7 = 3, %e A290557 a(2) = ( 13)_7 = 10, %e A290557 a(3) = ( 213)_7 = 108, %e A290557 a(4) = ( 6213)_7 = 2166, %e A290557 a(5) = (16213)_7 = 4567. %o A290557 (PARI) a(n) = truncate(sqrt(2+O(7^(n)))); \\ _Michel Marcus_, Aug 06 2017 %o A290557 (PARI) a(n) = lift(sqrt(2 + O(7^n))); \\ _Robert L. Brown_, Jun 16 2025 %Y A290557 Cf. A034945, A051277, A290559. %K A290557 nonn,easy %O A290557 0,2 %A A290557 _Seiichi Manyama_, Aug 05 2017