A047206 - OEIS

A047206

Numbers that are congruent to {1, 3, 4} mod 5.

1, 3, 4, 6, 8, 9, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 33, 34, 36, 38, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 68, 69, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 88, 89, 91, 93, 94, 96, 98, 99, 101, 103, 104, 106, 108

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OFFSET

1,2

LINKS

Table of n, a(n) for n=1..65.

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

FORMULA

G.f.: x*(1+2*x+x^2+x^3)/((1-x)^2*(1+x+x^2)).

a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.

a(n) = 1+(5*n)/3-(i*sqrt(3) * (-1/2+(i*sqrt(3))/2)^n)/9+(i*sqrt(3)* (-1/2-(i*sqrt(3))/2)^n)/9. - _Stephen Crowley_, Feb 11 2007

a(n) = floor((5*n-1)/3). - _Gary Detlefs_, May 14 2011

From _Wesley Ivan Hurt_, Jun 14 2016: (Start)

a(n) = (15*n-6-3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9.

a(3k) = 5k-1, a(3k-1) = 5k-2, a(3k-2) = 5k-4. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt((5-sqrt(5))/2)*Pi/5 + log(phi)/sqrt(5) - log(2)/5, where phi is the golden ratio (A001622). - _Amiram Eldar_, Apr 16 2023

E.g.f.: (9 + 3*exp(x)*(5*x - 2) - exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/9. - _Stefano Spezia_, Jun 22 2024

MAPLE

A047206:=n->(15*n-6-3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9: seq(A047206(n), n=1..100); # _Wesley Ivan Hurt_, Jun 14 2016

MATHEMATICA

Select[Range[0, 200], MemberQ[{1, 3, 4}, Mod[#, 5]] &] (* _Vladimir Joseph Stephan Orlovsky_, Feb 12 2012 *)

PROG

(Magma) [ n : n in [1..150] | n mod 5 in [1, 3, 4] ]; // _Vincenzo Librandi_, Mar 31 2011

(PARI) a(n)=(5*n-1)\3 \\ _Charles R Greathouse IV_, Jul 01 2013

CROSSREFS

Cf. A001622.

Sequence in context: A059535 A061402 A330066 * A187474 A081031 A285681

Adjacent sequences: A047203 A047204 A047205 * A047207 A047208 A047209

KEYWORD

nonn,easy

AUTHOR

_N. J. A. Sloane_

STATUS

approved