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A290557
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One of the two successive approximations up to 7^n for the 7-adic integer sqrt(2). These are the numbers congruent to 3 mod 7 (except for the initial 0).
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10
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0, 3, 10, 108, 2166, 4567, 38181, 155830, 1802916, 24862120, 266983762, 1961835256, 5916488742, 19757775943, 116646786350, 116646786350, 9611769806236, 42844700375837, 275475214363044, 6789129606004840, 75182500718243698, 154974767015855699
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OFFSET
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0,2
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COMMENTS
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x = ...216213,
x^2 = ...000002 = 2.
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LINKS
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FORMULA
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a(0) = 0 and a(1) = 3, a(n) = a(n-1) + (a(n-1)^2 - 2) mod 7^n for n > 1.
a(n) == 2*T(7^n, 3/2) (mod 7^n) == ((3 + sqrt(5))/2)^(7^n) + ((3 - sqrt(5))/2)^(7^n) (mod 7^n), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Dec 03 2022
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EXAMPLE
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a(1) = ( 3)_7 = 3,
a(2) = ( 13)_7 = 10,
a(3) = ( 213)_7 = 108,
a(4) = ( 6213)_7 = 2166,
a(5) = (16213)_7 = 4567.
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PROG
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(PARI) a(n) = truncate(sqrt(2+O(7^(n)))); \\ Michel Marcus, Aug 06 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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