A322217 Expansion of the 2-adic integer sqrt(17) that ends in 01.

1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0

Offset

0

Comments

Over the 2-adic integers there are 2 solutions to x^2 = 17, one ends in 01 and the other ends in 11. This sequence gives the former one. See A341538 for detailed information. - Jianing Song, Feb 13 2021

Links

Jianing Song, Table of n, a(n) for n = 0..1000

Formula

From Jianing Song, Feb 14 2021: (Start)

a(0) = 1, a(1) = 0; for n >= 2, a(n) = 0 if A341538(n)^2 - 17 is divisible by 2^(n+2), otherwise 1.

a(n) = 1 - A341540(n) for n >= 1.

For n >= 2, a(n) = (A341538(n+1) - A341538(n))/2^n. (End)

Example

sqrt(17) (2-adic) = ...110100010010000011111001100110011110100110010011011101001 = 1 + 2^3 + 2^5 + 2^6 + 2^7 + 2^9 + 2^10 + 2^13 + 2^16 + 2^17 + 2^20 + 2^22 + 2^23 + 2^24 + 2^25 + 2^28 + 2^29 + 2^32 + 2^33 + 2^36 + 2^37 + 2^38 + 2^39 + 2^40 + 2^46 + 2^49 + 2^53 + 2^55 + 2^56 + 2^58 + 2^63 + 2^66 + 2^67 + 2^68 + 2^74 + 2^75 + 2^77 + 2^79 + 2^80 + 2^83 + 2^86 + 2^94 + 2^96 + 2^98 + 2^99  + ...
The exponents of 2 in the above series begins:
[0, 3, 5, 6, 7, 9, 10, 13, 16, 17, 20, 22, 23, 24, 25, 28, 29, 32, 33, 36, 37, 38, 39, 40, 46, 49, 53, 55, 56, 58, 63, 66, 67, 68, 74, 75, 77, 79, 80, 83, 86, 94, 96, 98, 99, 101, 104, 105, 110, 112, 114, 115, 117, 119, 120, 122, 123, 125, 129, 130, 132, 136, 137, 138, 140, 143, 144, 145, 147, 150, 154, 158, 159, 160, 162, 163, 164, 165, 167, 168, 170, 174, 175, 180, 184, 185, 188, 189, 190, 192, 196, 198, ...].
The least positive real solution to the following power series in x with the same exponents:
sqrt(17) = 1 + x^3 + x^5 + x^6 + x^7 + x^9 + x^10 + x^13 + x^16 + x^17 + x^20 + x^22 + x^23 + x^24 + x^25 + x^28 + x^29 + ...
is x = 0.868455038691709167...

Prog

(PARI) /* Print the coefficients of powers of 2 in reverse order: */
Vecrev(binary(truncate( sqrt(17 + O(2^1001)) )))
(PARI) a(n) = truncate(sqrt(17+O(2^(n+2))))\2^n \\ Jianing Song, Feb 13 2021

Crossrefs

Cf. A341540, A341538 (successive approximations of the associated 2-adic square root of 17), A318962, A318963 (expansion of sqrt(-7)).

Equals A341753^2 = A341754^2.

Sequence in context: A011749 A277450 A188578 * A104105 A295889 A347523

Adjacent sequences: A322214 A322215 A322216 * A322218 A322219 A322220

Keyword

nonn,base

Author

Paul D. Hanna, Dec 14 2018

Extensions

Name edited by Jianing Song, Feb 13 2021

Status

approved