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A210850 Digits of one of the two 5-adic integers sqrt(-1). 24
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

If n>0, a(n) == A210848(n) (mod 5), since A048898(n) == 2 (mod 5). - Álvar Ibeas, Feb 21 2017

If a(n)=0 then A048899(n+1) and A048899(n) coincide.

a(n) + A210851(n) = 4 for n >= 1. - Robert Israel, Mar 04 2016

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

FORMULA

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(a(k)*5^k, k=0..n), n>=0.

EXAMPLE

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(a(k)*5^k, k=0..7) = 280182.

MAPLE

R:= select(t -> padic:-ratvaluep(t, 1)=2, [padic:-rootp(x^2+1, 5, 10001)]):

op([1, 1, 3], R); # Robert Israel, Mar 04 2016

PROG

(PARI) a(n) = truncate(sqrt(-1+O(5^(n+1))))\5^n; \\ Michel Marcus, Mar 05 2016

CROSSREFS

Cf. A048898, A210848, A048899, A210849, A210851.

Sequence in context: A329382 A322826 A133117 * A051276 A226212 A233439

Adjacent sequences:  A210847 A210848 A210849 * A210851 A210852 A210853

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Apr 30 2012

STATUS

approved

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Last modified February 19 22:04 EST 2020. Contains 332060 sequences. (Running on oeis4.)