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A341754
Expansion of the 2-adic integer 17^(1/4) that ends in 11.
4
1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1
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 latter one. See A341752 for detailed information.
LINKS
FORMULA
a(0) = 1, a(1) = 0; for n >= 2, a(n) = 0 if A341752(n)^4 - 17 is divisible by 2^(n+3), otherwise 1.
a(n) = 1 - A341753(n) for n >= 1.
For n >= 2, a(n) = (A341752(n+1) - A341752(n))/2^n.
EXAMPLE
If x = ...00100010001100111011000011010001001010011, 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. A341752 (successive approximations of the 2-adic fourth root of 17), A322217.
Approximations of p-adic fourth-power roots:
A341753, this sequence (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: A114915 A360120 A361022 * A074711 A004585 A319448
KEYWORD
nonn,base
AUTHOR
Jianing Song, Feb 18 2021
STATUS
approved