|
|
A290559
|
|
One of the two successive approximations up to 7^n for the 7-adic integer sqrt(2). These are the numbers congruent to 4 mod 7 (except for the initial 0).
|
|
10
|
|
|
0, 4, 39, 235, 235, 12240, 79468, 667713, 3961885, 15491487, 15491487, 15491487, 7924798459, 77131234464, 561576286499, 4630914723593, 23621160763365, 189785813611370, 1352938383547405, 4609765579368303, 4609765579368303, 403571097067428308
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
x = ...450454,
x^2 = ...000002 = 2.
|
|
LINKS
|
|
|
FORMULA
|
a(0) = 0 and a(1) = 4, a(n) = a(n-1) + 6 * (a(n-1)^2 - 2) mod 7^n for n > 1.
a(n) == 2*T(7^n, 2) (mod 7^n) == (2 + sqrt(3))^(7^n) + (2 - sqrt(3))^(7^n) (mod 7^n), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Dec 03 2022
|
|
EXAMPLE
|
a(1) = ( 4)_7 = 4,
a(2) = ( 54)_7 = 39,
a(3) = ( 454)_7 = 235,
a(4) = ( 454)_7 = 235,
a(5) = (50454)_7 = 12240.
|
|
PROG
|
(PARI) a(n) = if (n==0, 0, 7^n - truncate(sqrt(2+O(7^n)))); \\ Michel Marcus, Aug 06 2017
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|