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

0,1

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.

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

Example

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

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

Prog

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

Crossrefs

Cf. A034945, A290558.

Sequence in context: A049919 A246432 A112571 * A300908 A080818 A334354

Adjacent sequences: A051274 A051275 A051276 * A051278 A051279 A051280

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

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

Status

approved