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A269592
Digits of one of the two 5-adic integers sqrt(-4). Here the ones related to A269590.
16
4, 2, 4, 2, 1, 4, 0, 2, 1, 1, 0, 0, 1, 3, 3, 1, 0, 4, 1, 3, 2, 4, 1, 3, 3, 4, 3, 3, 3, 3, 2, 1, 3, 3, 3, 3, 0, 1, 2, 2, 1, 2, 0, 0, 4, 1, 3, 0, 4, 1, 1, 3, 4, 3, 1, 1, 2, 1, 1, 1, 0, 0, 1, 3, 1, 3
OFFSET
0,1
COMMENTS
This is the scaled first difference sequence of A269590.
The digits of the other 5-adic integer sqrt(-4), are given in A269591. See also A268922 for the two 5-adic numbers -u and u.
a(n) is the unique solution of the linear congruence 2*A269590(n)*a(n) + A269594(n) == 0 (mod 5), n>=1. Therefore only the values 0, 1, 2, 3 and 4 appear. See the Nagell reference given in A268922, eq. (6) on p. 86, adapted to this case. a(0) = 4 follows from the formula given below.
FORMULA
a(n) = (b(n+1) - b(n))/5^n, n>=0, with b(n) = A269590(n), n >= 0.
a(n) = -A269594(n)*(2*A269590(n))^3 (mod 5), n >= 1. Solution of the linear congruence see, e.g., Nagell, Theorem 38 pp. 77-78.
A269590(n+1) = sum(a(k)*5^k, k=0..n), n>=0.
a(n) = 4 - A269591(n) if n > 0 and a(0) = 5 - A269591(0) = 5-4 = 1. - Michel Marcus, Mar 31 2016
EXAMPLE
a(4) = -212*(2*364)^3 (mod 5) = -2*(2*(-1))^3 (mod 5) = 1.
PROG
(PARI) a(n) = truncate(-sqrt(-4+O(5^(n+1))))\5^n; \\ Michel Marcus, Mar 04 2016
CROSSREFS
Cf. A269590, A269591 (companion), A269594.
Sequence in context: A016514 A224360 A375087 * A105256 A064127 A178253
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Mar 02 2016
STATUS
approved