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A210848 a(n) = (A048898(n)^2 + 1)/5^n, n >= 0. 11
1, 1, 2, 26, 53, 1354, 13562, 26858, 200965, 40193, 3859882, 13496122, 62298370, 12459674, 4106065226, 4044371993, 69072101242, 218014644394, 3137550252170, 627510050434, 66696011833378, 280704828874769 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is integer (nonnegative) because b(n):=A048898(n) satisfies b(n)^2 + 1 == 0 (mod 5^n), n>=0. The solution of this congruence for n>=1, which satisfies also b(n) == 2 (mod 5), is b(n) = 2^(5^(n-1)) (mod 5^n), but this is inconvenient for computing b(n) for large n. Instead one can use the b(n) recurrence which follows immediately, and this is given in the formula field below. To prove that the given b(n) formula solves the first congruence one can analyze the binomial expansion of (5 - 1)^(5^(n-1)) + 1 and show that it is 0 (mod 5^n)) term by term. The second congruence reduces to b(n) == 2^(5^(n-1)) (mod 5) which follows for n>=1 by induction. Because b(n) = 5^n - A048899(n) one could also use the result A048899(n) == 3 (mod 5), once this has been proved.

The fact that X^2 + 1 == 0 (mod 5^n) has precisely two solutions for each n>=1, called x(n) and y(n), follows from the fact that X^2 + 1 == 0 (mod 5) has the two simple roots x(1) = 2 and y(1) = 3, and a theorem, given, e.g., in the Nagell reference as Theorem 50 on p. 87. From that same theorem, it also follows that one can choose all x(n) == 2 (mod 5) and all y(n) == 3 (mod 5).

REFERENCES

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

LINKS

Table of n, a(n) for n=0..21.

FORMULA

a(n) = (b(n)^2+1)/5^n, n>=0, with b(n) = A048898(n) given by the recurrence b(n) = b(n-1)^5 (mod 5^n), n>=2, b(0):=0, b(1)=2 (see the Mathematica Program by Jean-Fran├žois Alcover at A048898).

EXAMPLE

a(0) = 1/1 = 1.

a(3) = (57^2 + 1)/5^3 = 26 (b(3) = 7^5 (mod 5^3) = 57).

MAPLE

b:=proc(n) option remember: if n=0 then 0 elif n=1 then 2

else modp(b(n-1)^5, 5^n) fi: end proc:

[seq((b(n)^2+1)/5^n, n=0..29)];

CROSSREFS

Cf. A048898, A048899, A210849 (companion sequence).

Sequence in context: A001772 A199206 A132861 * A247957 A152997 A229573

Adjacent sequences:  A210845 A210846 A210847 * A210849 A210850 A210851

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Apr 28 2012

STATUS

approved

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Last modified September 19 23:31 EDT 2019. Contains 327207 sequences. (Running on oeis4.)