

A176514


Period 6: repeat [3, 1, 1, 3, 2, 1].


1



3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 1, 3, 2, 1
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OFFSET

1,1


COMMENTS

3/n expressed as Egyptian fractions with 3 unit fractions: 3/n = 1/x + 1/y + 1/z, all integers, n > 0 and x < y < z.
I have introduced a variable t, an integer that varies with n in accordance with a periodic sequence of 6 terms: [3, 1, 1, 3, 2, 1] starting at n = 0, although 0 is outside the defined interval.
x, the denominator of the first unit fraction, varies with n in accordance with the sequence:
[(1), 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5,..]; x_0 = x_1 = x_2 = 1. Then x_3 = x_4 = x_5 = 2 and so on, increasing the value by 1 for every 3 terms.
y is calculated by:
y = (nx + t)/2 for n = 1, 4, 7, 10,.. etc.
y = (nx + t)/1 = nx + t for n = 2, 5, 8, 11,.. etc.
y = (nx + t)/t for n = 3, 6, 9, 12,.. etc.
z is calculated by: z = nxy/t for all n > 0
This algorithm produces the "first" Egyptian fraction of each type that has 3 unit fractions.
By "first" I indicate the Egyptian fraction that otherwise would be arrived at by employing Fibonacci's greedy algorithm.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).


FORMULA

a(n) = (1/90)*{26*(n mod 6)+26*[(n+1) mod 6]19*[(n+2) mod 6]+11*[(n+3) mod 6]+41*[(n+4) mod 6]19*[(n+5) mod 6]}.  Paolo P. Lava, Apr 21 2010
G.f.: ( 3+x+x^2+3*x^3+2*x^4+x^5 ) / ( (1x)*(1+x)*(1+x+x^2)*(x^2x+1) ).  R. J. Mathar, Oct 08 2011
a(n) = 11/6 cos(Pi*n/3)/6 sqrt(3)*sin(Pi*n/3)/6 +7*cos(2*Pi*n/3)/6 +sqrt(3)*sin(2*Pi*n/3)/6 +(1)^n/6.  R. J. Mathar, Oct 08 2011
a(n) = a(n6) for n>6.  Wesley Ivan Hurt, Jun 18 2016


MAPLE

A176514:=n>[3, 1, 1, 3, 2, 1][(n mod 6)+1]: seq(A176514(n), n=0..100); # Wesley Ivan Hurt, Jun 18 2016


MATHEMATICA

LinearRecurrence[{0, 0, 0, 0, 0, 1}, {3, 1, 1, 3, 2, 1}, 96] (* Ray Chandler, Aug 26 2015 *)
PadRight[{}, 100, {3, 1, 1, 3, 2, 1}] (* Vincenzo Librandi, Jun 19 2016 *)


PROG

(MAGMA) &cat[[3, 1, 1, 3, 2, 1]^^20]; // Wesley Ivan Hurt, Jun 18 2016
(PARI) a(n)=[1, 3, 1, 1, 3, 2][n%6+1] \\ Charles R Greathouse IV, Jul 17 2016


CROSSREFS

Sequence in context: A106749 A225224 A140216 * A238559 A077196 A023142
Adjacent sequences: A176511 A176512 A176513 * A176515 A176516 A176517


KEYWORD

nonn,easy


AUTHOR

Egil Edborg (egil.edborg(AT)ebnett.no), Apr 19 2010


STATUS

approved



