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%I #44 Apr 16 2023 08:38:23
%S 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,
%T 43,44,46,48,49,51,53,54,56,58,59,61,63,64,66,68,69,71,73,74,76,78,79,
%U 81,83,84,86,88,89,91,93,94,96,98,99,101,103,104,106,108
%N Numbers that are congruent to {1, 3, 4} mod 5.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F G.f.: (1+2*x+x^2+x^3)/((1-x)^2*(1+x+x^2)).
%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
%F 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
%F a(n) = floor((5*n-1)/3). - _Gary Detlefs_, May 14 2011
%F From _Wesley Ivan Hurt_, Jun 14 2016: (Start)
%F a(n) = (15*n-6-3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9.
%F a(3k) = 5k-1, a(3k-1) = 5k-2, a(3k-2) = 5k-4. (End)
%F 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
%p 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
%t Select[Range[0, 200], MemberQ[{1, 3, 4}, Mod[#, 5]] &] (* _Vladimir Joseph Stephan Orlovsky_, Feb 12 2012 *)
%o (Magma) [ n : n in [1..150] | n mod 5 in [1, 3, 4] ]; // _Vincenzo Librandi_, Mar 31 2011
%o (PARI) a(n)=(5*n-1)\3 \\ _Charles R Greathouse IV_, Jul 01 2013
%Y Cf. A001622.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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