A032793 Numbers that are congruent to {1, 2, 4} mod 5.
1, 2, 4, 6, 7, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 32, 34, 36, 37, 39, 41, 42, 44, 46, 47, 49, 51, 52, 54, 56, 57, 59, 61, 62, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 81, 82, 84, 86, 87, 89, 91, 92, 94, 96, 97, 99, 101, 102, 104, 106, 107, 109
Offset: 1
Links
- Guenther Schrack, Table of n, a(n) for n = 1..10000
- Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Programs
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GAP
a:=[1,2,4,6];; for n in [5..100] do a[n]:=a[n-1]+a[n-3]-a[n-4]; od; a; # G. C. Greubel, Nov 06 2019
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Magma
[ n: n in [0..120] | n mod 5 in {1, 2, 4} ]; // Vincenzo Librandi, Dec 29 2010 -
Maple
A032793:=n->(15*n-9+2*sqrt(3)*sin(2*n*Pi/3))/9: seq(A032793(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016
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Mathematica
Select[Range[0, 200], MemberQ[{1, 2, 4}, Mod[#, 5]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *) LinearRecurrence[{1,0,1,-1},{1,2,4,6},90] (* Harvey P. Dale, May 20 2019 *) -
PARI
a(n)=n\3*5+[-1,1,2][n%3+1] \\ Charles R Greathouse IV, Jan 18 2012
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Sage
[(15*n - 9 + 2*sqrt(3)*sin(2*n*pi/3))/9 for n in (1..100)] # G. C. Greubel, Nov 06 2019
Formula
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = floor((5*n-2)/3). - Gary Detlefs, May 14 2011
G.f.: x*(1+x+2*x^2+x^3)/((1+x+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = (15*n - 9 + 2*sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 5k - 1, a(3k-1) = 5k - 3, a(3k-2) = 5k - 4. (End)
E.g.f.: (9 + 3*(5*x - 3)*exp(x) + 2*sqrt(3)*sin(sqrt(3)*x/2)*(cosh(x/2) - sinh(x/2)))/9. - Ilya Gutkovskiy, Jun 14 2016
From Guenther Schrack, Oct 31 2019: (Start)
a(n) = a(n-3) + 5 with a(1) = 1, a(2) = 2, a(3) = 4 for n > 3.
a(n) = (15*n - 9 + (w^(2*n) - w^n)*(1 + 2*w))/9 where w = (-1 + sqrt(-3))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(10-2*sqrt(5))*Pi/10 - log(phi)/sqrt(5) + log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 16 2023
Extensions
Better description from Michael Somos, Jun 08 2000