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A290803
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One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-3). These are the numbers congruent to 2 mod 7 (except for the initial 0).
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10
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0, 2, 37, 37, 2095, 14100, 47714, 165363, 988906, 29812911, 271934553, 1401835549, 9311142521, 36993716923, 133882727330, 2846775018726, 12341898038612, 145273620317016, 1308426190253051, 1308426190253051, 35505111746372480, 354674176936820484
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OFFSET
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0,2
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COMMENTS
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x = ...256052,
x^2 = ...666664 = -3.
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LINKS
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FORMULA
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a(0) = 0 and a(1) = 2, a(n) = a(n-1) + 5 * (a(n-1)^2 + 3) mod 7^n for n > 1.
a(n) = L(7^n,2) (mod 7^n) = ( (1 + sqrt(2))^(7^n) + (1 - sqrt(2))^(7^n) ) (mod 7^n), where L(n,x) denotes the n-th Lucas polynomial of A114525. - Peter Bala, Nov 28 2022
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EXAMPLE
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a(1) = 2_7 = 2,
a(2) = 52_7 = 37,
a(3) = 52_7 = 37,
a(4) = 6052_7 = 2095,
a(5) = 56052_7 = 14100.
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PROG
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(PARI) a(n) = if (n, truncate(sqrt(-3+O(7^(n)))), 0)
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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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