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A341753
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Expansion of the 2-adic integer 17^(1/4) that ends in 01.
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4
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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
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OFFSET
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0
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COMMENTS
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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.
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LINKS
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Jianing Song, Table of n, a(n) for n = 0..1000
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FORMULA
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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.
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EXAMPLE
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If x = ...11011101110011000100111100101110110101101, then x^2 = ...1111001100110011110100110010011011101001 = A322217, x^4 = 10001_2 = 17.
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PROG
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(PARI) a(n) = truncate(sqrtn(17+O(2^(n+3)), 4))\2^n
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CROSSREFS
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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
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KEYWORD
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nonn,base
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AUTHOR
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Jianing Song, Feb 18 2021
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STATUS
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approved
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