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A341603
Expansion of the 2-adic integer sqrt(-3/5) that ends in 11.
4
1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1
OFFSET
0
COMMENTS
Over the 2-adic integers there are 2 solutions to 5*x^2 + 3 = 0, one ends in 01 and the other ends in 11. This sequence gives the latter one. See A341601 for detailed information.
This constant may be used to represent one of the two primitive 6th roots of unity, namely one of the two roots of x^2 - x + 1 = 0 in Q_2(sqrt(5)), the unique unramified quatratic extension of the 2-adic field: if x = (1 + A341603*sqrt(5))/2, then x^2 = (-1 + A341603*sqrt(5))/2, x^3 = -1, x^4 = (-1 + A341602*sqrt(5))/2, x^5 = (1 + A341602*sqrt(5))/2 and x^6 = 1.
In the ring of 2-adic integers the sequence {Fibonacci(4^n)} converges to this constant. For example, Fibonacci(4^10) reduced modulo 4^10 = 651835 = 10011111001000111011 (binary representation). Reading the binary digits from right to left gives the first 20 terms of this sequence. - Peter Bala, Nov 22 2022
FORMULA
a(0) = 1, a(1) = 0; for n >= 2, a(n) = 0 if 5*A341601(n)^2 + 3 is divisible by 2^(n+2), otherwise 1.
a(n) = 1 - A341602(n) for n >= 1.
For n >= 2, a(n) = (A341601(n+1) - A341601(n))/2^n.
EXAMPLE
If v = ...00001100110000000011010011111001000111011, then v^2 = ...1001100110011001100110011001100110011001 = -3/5. Furthermore, let x = (1 + v*sqrt(5))/2, then x^2 = (-1 + v*sqrt(5))/2, x^3 = -1, x^4 = (-1 - v*sqrt(5))/2, x^5 = (1 - v*sqrt(5))/2 and x^6 = 1.
PROG
(PARI) a(n) = if(n==1, 1, truncate(sqrt(-3/5+O(2^(n+2))))\2^n)
CROSSREFS
Cf. A145231, A341602, A341601 (successive approximations of the associated 2-adic square root of -3/5), A318962, A318963 (expansion of sqrt(-7)), A322217, A341540 (expansion of sqrt(17)).
Sequence in context: A014114 A014219 A065828 * A190897 A176329 A305994
KEYWORD
nonn,base,easy
AUTHOR
Jianing Song, Feb 16 2021
STATUS
approved