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A290804
One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-3). These are the numbers congruent to 5 mod 7 (except for the initial 0).
10
0, 5, 12, 306, 306, 2707, 69935, 658180, 4775895, 10540696, 10540696, 575491194, 4530144680, 59895293484, 544340345519, 1900786491217, 20891032530989, 87356893670191, 319987407657398, 10090468995120092, 44287154551239521, 203871687146463523
OFFSET
0,2
COMMENTS
x = ...410615,
x^2 = ...666664 = -3.
FORMULA
a(0) = 0 and a(1) = 5, a(n) = a(n-1) + 2 * (a(n-1)^2 + 3) mod 7^n for n > 1.
If n > 0, a(n) = 7^n - A290803(n).
a(n) = L(7^n,5) (mod 7^n) = ( ((5 + sqrt(29))/2)^(7^n) + ((5 - sqrt(29))/2)^(7^n) ) (mod 7^n), where L(n,x) denotes the n-th Lucas polynomial of A114525. - Peter Bala, Nov 28 2022
EXAMPLE
a(1) = 5_7 = 5,
a(2) = 15_7 = 12,
a(3) = 615_7 = 306,
a(4) = 615_7 = 306,
a(5) = 10615_7 = 2707.
PROG
(PARI) a(n) = if (n, 7^n - truncate(sqrt(-3+O(7^(n)))), 0)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Aug 11 2017
STATUS
approved