A051277 - OEIS

%I #29 Dec 04 2022 13:06:11

%S 3,1,2,6,1,2,1,2,4,6,6,2,1,1,0,2,1,1,4,6,1,3,2,6,6,3,5,5,6,3,4,5,0,1,

%T 6,3,0,4,6,2,4,4,6,4,2,4,4,2,6,1,3,4,1,3,1,4,2,6,6,0,3,5,5,1,1,2,0,6,

%U 6,1,1,2,4,4,4,2,3,6,6,3,6,1,4,4,2,2,1,3

%N Coefficients in 7-adic expansion of sqrt(2).

%D Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 76.

%H Seiichi Manyama, <a href="/A051277/b051277.txt">Table of n, a(n) for n = 0..10000</a>

%H Peter Bala, <a href="/A051277/a051277.pdf">Using Chebyshev polynomials to find the p-adic square roots of 2 and 3</a>, Dec 2022.

%F Equals the 7-adic limit as n -> oo of 2*T(7^n,3/2) = the 7-adic limit as n -> oo of ((3 + sqrt(5))/2)^(7^n) + ((3 - sqrt(5))/2)^(7^n), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - _Peter Bala_, Nov 20 2022

%e 3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + ...

%p  t := proc(n) option remember; if n = 1 then 3 else irem(t(n-1)^7 - 7*t(n-1)^5 + 14*t(n-1)^3 - 7*t(n-1), 7^n) end if; end:

%p convert(t(100), base, 7); # _Peter Bala_, Nov 20 2022

%o (PARI) Vecrev(digits(lift(sqrt(2+O(7^99))),7)) \\ _Joerg Arndt_, Aug 05 2017

%Y Cf. A034945, A290558.

%K nonn,easy,nice

%O 0,1

%A _N. J. A. Sloane_

%E Missing terms=0 inserted by _Seiichi Manyama_, Aug 04 2017