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A047206 Numbers that are congruent to {1, 3, 4} mod 5.

%I A047206 #54 Mar 12 2025 08:21:04
%S A047206 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 A047206 43,44,46,48,49,51,53,54,56,58,59,61,63,64,66,68,69,71,73,74,76,78,79,
%U A047206 81,83,84,86,88,89,91,93,94,96,98,99,101,103,104,106,108
%N A047206 Numbers that are congruent to {1, 3, 4} mod 5.
%C A047206 a(n) is the maximum number of heads achievable in the game of blet with 2*n coins. See A075274 and A381812. - _Pontus von Brömssen_, Mar 09 2025
%H A047206 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F A047206 G.f.: x*(1+2*x+x^2+x^3)/((1-x)^2*(1+x+x^2)).
%F A047206 a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
%F A047206 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 A047206 a(n) = floor((5*n-1)/3). - _Gary Detlefs_, May 14 2011
%F A047206 From _Wesley Ivan Hurt_, Jun 14 2016: (Start)
%F A047206 a(n) = (15*n-6-3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9.
%F A047206 a(3k) = 5k-1, a(3k-1) = 5k-2, a(3k-2) = 5k-4. (End)
%F A047206 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
%F A047206 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
%p A047206 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 A047206 Select[Range[0, 200], MemberQ[{1, 3, 4}, Mod[#, 5]] &] (* _Vladimir Joseph Stephan Orlovsky_, Feb 12 2012 *)
%o A047206 (Magma) [ n : n in [1..150] | n mod 5 in [1, 3, 4] ]; // _Vincenzo Librandi_, Mar 31 2011
%o A047206 (PARI) a(n)=(5*n-1)\3 \\ _Charles R Greathouse IV_, Jul 01 2013
%Y A047206 Cf. A001622, A075274, A381812.
%K A047206 nonn,easy
%O A047206 1,2
%A A047206 _N. J. A. Sloane_