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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 (list; graph; refs; listen; history; text; internal format)
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 ) / ( (1-x)*(1+x)*(1+x+x^2)*(x^2-x+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(n-6) 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

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Last modified June 13 07:19 EDT 2021. Contains 344981 sequences. (Running on oeis4.)