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A210850
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Digits of one of the two 5-adic integers sqrt(-1).
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33
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2, 1, 2, 1, 3, 4, 2, 3, 0, 3, 2, 2, 0, 4, 1, 3, 2, 4, 0, 4, 3, 4, 0, 4, 1, 2, 4, 1, 4, 1, 1, 3, 1, 4, 1, 4, 2, 0, 1, 1, 3, 3, 2, 2, 4, 0, 4, 2, 4, 0, 3, 1, 2, 4, 0, 3, 3, 0, 3, 0, 0, 0, 3, 1, 3, 1, 1, 0, 3, 0, 0, 3, 4, 1, 3, 3, 3, 4, 0, 2, 2, 0, 2, 0, 1, 0, 4, 1, 1, 4, 4, 2, 1, 0, 2, 0, 0, 3, 0, 4
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OFFSET
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0,1
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COMMENTS
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See A048898 for the successive approximations to this 5-adic integer, called there u.
The digits of -u, the other 5-adic integer sqrt(-1), are given in A210851.
a(n) is the unique solution of the linear congruence 2*A048898(n)*a(n) + A210848(n) == 0 (mod 5), n>=1. Therefore only the values 0, 1, 2, 3 and 4 appear. See the Nagell reference given in A210848, eq. (6) on p. 86 adapted to this case. a(0)=2 follows from the formula given below.
With a(0) = 1, this is the digits of one of the four 4th root of -4 in the ring of 5-adic integers, the one that is congruent to 1 modulo 5.
With a(0) = 3, this is the digits of one of the four 4th root of -4 in the ring of 5-adic integers, the one that is congruent to 3 modulo 5. (End)
This square root of -1 in the 5-adic integers is equal to the 5-adic limit of the sequence {L(5^n,2)}, where L(n,x) denotes the n-th Lucas polynomial, the n-th row polynomial of A114525. - Peter Bala, Dec 02 2022
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LINKS
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FORMULA
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a(n) = (b(n+1) - b(n))/5^n, n >= 0, with b(n):=A048898(n) computed from its recurrence. A Maple program for b(n) is given there.
A048898(n+1) = Sum_{k=0..n} a(k)*5^k, n >= 0.
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EXAMPLE
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a(4) = 3 because 2*182*3 + 53 = 1145 == 0 (mod 5).
A048898(5) = 2057 = 2*5^0 + 1*5^1 + 2*5^2 + 1*5^3 + 3*5^4.
a(8) = 0, therefore A048898(9) = A048898(8) = Sum_{k=0..7} a(k)*5^k = 280182.
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MAPLE
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R:= select(t -> padic:-ratvaluep(t, 1)=2, [padic:-rootp(x^2+1, 5, 10001)]):
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MATHEMATICA
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Table[Floor[First@Select[PowerModList[-1, 1/2, 5^(k+1)], Mod[#, 5]==2&]/5^k], {k, 0, 99}] (* Giorgos Kalogeropoulos, Feb 28 2023 *)
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PROG
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(PARI) a(n) = truncate(sqrt(-1+O(5^(n+1))))\5^n; \\ Michel Marcus, Mar 05 2016
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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