A051277 Coefficients in 7-adic expansion of sqrt(2).

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, 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, 6, 1, 1, 2, 4, 4, 4, 2, 3, 6, 6, 3, 6, 1, 4, 4, 2, 2, 1, 3

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Examples

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

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Peter Bala, Using Chebyshev polynomials to find the p-adic square roots of 2 and 3, Dec 2022.

Crossrefs

Cf. A034945, A290558.

Programs

Maple

 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:
convert(t(100), base, 7); # Peter Bala, Nov 20 2022

PARI

Vecrev(digits(lift(sqrt(2+O(7^99))),7)) \\ Joerg Arndt, Aug 05 2017

Formula

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

Extensions

Missing terms=0 inserted by Seiichi Manyama, Aug 04 2017