A341753 Expansion of the 2-adic integer 17^(1/4) that ends in 01.

1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0

Offset

0

Comments

Also square root of A322217.

Over the 2-adic integers, for k == 1 (mod 16), there are 2 solutions to x^4 = k, one ends in 01 and the other ends in 11. This sequence gives the former one. See A341751 for detailed information.

Links

Jianing Song, Table of n, a(n) for n = 0..1000

Formula

a(0) = 1, a(1) = 0; for n >= 2, a(n) = 0 if A341751(n)^4 - 17 is divisible by 2^(n+3), otherwise 1.

a(n) = 1 - A341754(n) for n >= 1.

For n >= 2, a(n) = (A341751(n+1) - A341751(n))/2^n.

A341753^2 = A322217.

Example

If x = ...11011101110011000100111100101110110101101, then x^2 = ...1111001100110011110100110010011011101001 = A322217, x^4 = 10001_2 = 17.

Prog

(PARI) a(n) = truncate(sqrtn(17+O(2^(n+3)), 4))\2^n

Crossrefs

Cf. A341751 (successive approximations of the 2-adic fourth root of 17), A322217.

Approximations of p-adic fourth-power roots:

this sequence, A341754 (2-adic, 17^(1/4));

A325489, A325490, A325491, A325492 (5-adic, 6^(1/4));

A324085, A324086, A324087, A324153 (13-adic, 3^(1/4)).

Sequence in context: A125144 A115198 A005614 * A267605 A319843 A309847

Adjacent sequences: A341750 A341751 A341752 * A341754 A341755 A341756

Keyword

nonn,base

Author

Jianing Song, Feb 18 2021

Status

approved