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A157810 Period 4: repeat [2, 1, 3, 2]. 4
2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let V be the set of primes p for which {(3^n-1)/(2^n-1) such that n is a natural number} is contained in Z(p) contained in Q denote the localization of the integral domain Z at the prime ideal (p); that is, the subring of Q consisting of the rational numbers with denominators prime to p. (a) Characterize the set V. (b) subproblem about Wieferich primes A001220. (c) Show that, for every p and element of V, the map N -> F_p given by n -> phi_p ((3^n-1)/(2^n-1}) is periodic. For example, 5 is an element of V, and the corresponding map N > F_5 is 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, ... .

Continued fraction expansion of (7+sqrt(93))/6. - Klaus Brockhaus, Apr 30 2010

LINKS

Table of n, a(n) for n=1..92.

Vesselin Dimitrov, Problem S07 - 4 (Corrected). Harvard College Mathematical Review, Vol. 1, No. 2, Fall 2007.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

FORMULA

a(n) = (1/12)*{4*[(n-1) mod 4]+7*(n mod 4)-2*[(n+1) mod 4]+7*[(n+2) mod 4]}. - Paolo P. Lava, Mar 17 2009

a(n) = a(n-4) for n>4. G.f.: x*(2*x^3+3*x^2+x+2) / ((1-x)*(x+1)*(x^2+1)). - Colin Barker, Jun 20 2014

a(n) = 2-(-1)^n/2+(-1)^ceiling(n/2)/2. - Wesley Ivan Hurt, Jun 23 2014

a(n) = (4 + cos(n*Pi/2) - cos(n*Pi) - sin(n*Pi/2) - I*sin(n*Pi))/2.

MAPLE

A157810:=n->2 - (-1)^n/2 + (-1)^ceil(n/2)/2; seq(A157810(n), n=1..100); # Wesley Ivan Hurt, Jun 23 2014

MATHEMATICA

ContinuedFraction[(7+Sqrt[93])/6, 100] (* Harvey P. Dale, Jun 28 2012 *)

CoefficientList[Series[-(2*x^3 + 3*x^2 + x + 2)/((x - 1)*(x + 1)*(x^2 + 1)), {x, 0, 60}], x] (* Wesley Ivan Hurt, Jun 22 2014 *)

PROG

(PARI) Vec(-x*(2*x^3+3*x^2+x+2)/((x-1)*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Jun 20 2014

(MAGMA) [2 - (-1)^n/2 + (-1)^Ceiling(n/2)/2 : n in [1..100]]; // Wesley Ivan Hurt, Jun 23 2014

CROSSREFS

Cf. A001220.

Sequence in context: A324923 A126303 A306467 * A072339 A261337 A260088

Adjacent sequences:  A157807 A157808 A157809 * A157811 A157812 A157813

KEYWORD

nonn,easy

AUTHOR

Jonathan Vos Post, Mar 07 2009

EXTENSIONS

Simpler definition from Wesley Ivan Hurt, Jul 07 2014

STATUS

approved

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Last modified June 20 19:36 EDT 2019. Contains 324234 sequences. (Running on oeis4.)