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
Offset: 1
Links
- Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Programs
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Magma
[ n : n in [1..150] | n mod 5 in [1, 3, 4] ]; // Vincenzo Librandi, Mar 31 2011
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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
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Mathematica
Select[Range[0, 200], MemberQ[{1, 3, 4}, Mod[#, 5]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *) -
PARI
a(n)=(5*n-1)\3 \\ Charles R Greathouse IV, Jul 01 2013
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
Comments