A047212 - OEIS

%I #51 Sep 08 2022 08:44:56

%S 0,2,4,5,7,9,10,12,14,15,17,19,20,22,24,25,27,29,30,32,34,35,37,39,40,

%T 42,44,45,47,49,50,52,54,55,57,59,60,62,64,65,67,69,70,72,74,75,77,79,

%U 80,82,84,85,87,89,90,92,94,95,97,99,100,102,104,105,107

%N Numbers that are congruent to {0, 2, 4} mod 5.

%C Also numbers k such that k*(k+1)*(k+3) is divisible by 5. - _Bruno Berselli_, Dec 28 2017

%H Charles R Greathouse IV, <a href="/A047212/b047212.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

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

%F G.f.: x^2*(2 + 2*x + x^2)/((1 - x)^2*(1 + x + x^2)). - _Bruno Berselli_, Mar 31 2011

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

%F a(n) = n + ceiling(2*(n-1)/3) - 1. - _Arkadiusz Wesolowski_, Sep 18 2012

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

%F a(n) = (15*n - 12 + 3*cos(2*n*Pi/3) - sqrt(3)*sin(2*n*Pi/3))/9.

%F a(3*k) = 5*k-1, a(3*k-1) = 5*k-3, a(3*k-2) = 5*k-5. (End)

%F Sum_{n>=2} (-1)^n/a(n) = log(2)/5 + arccosh(7/2)/(2*sqrt(5)) - sqrt(1-2*sqrt(5)/5)*Pi/5. - _Amiram Eldar_, Dec 10 2021

%p A047212:=n->floor((5*n-3)/3); seq(A047212(n), n=1..100); # _Wesley Ivan Hurt_, Nov 25 2013

%t Select[Range[0, 200], MemberQ[{0, 2, 4}, Mod[#, 5]] &] (* _Vladimir Joseph Stephan Orlovsky_, Feb 12 2012 *)

%o (Magma) [n : n in [0..140] | n mod 5 in [0, 2, 4]]; // _Vincenzo Librandi_, Mar 31 2011

%o (Magma) &cat[[n, n+2, n+4]: n in [0..90 by 5]]; // _Bruno Berselli_, Mar 31 2011

%o (PARI) a(n)=n\3*5+[-1,0,2][n%3+1] \\ _Charles R Greathouse IV_, Mar 29 2012

%Y Cf. A047209, A047211.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_