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Numbers that are congruent to {1, 2, 4} mod 5.
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%I #55 Dec 30 2023 23:44:57

%S 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,

%T 42,44,46,47,49,51,52,54,56,57,59,61,62,64,66,67,69,71,72,74,76,77,79,

%U 81,82,84,86,87,89,91,92,94,96,97,99,101,102,104,106,107,109

%N Numbers that are congruent to {1, 2, 4} mod 5.

%H Guenther Schrack, <a href="/A032793/b032793.txt">Table of n, a(n) for n = 1..10000</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 a(n) = floor((5*n-2)/3). - _Gary Detlefs_, May 14 2011

%F G.f.: x*(1+x+2*x^2+x^3)/((1+x+x^2)*(1-x)^2). - _R. J. Mathar_, Oct 08 2011

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

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

%F a(3k) = 5k - 1, a(3k-1) = 5k - 3, a(3k-2) = 5k - 4. (End)

%F 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

%F From _Guenther Schrack_, Oct 31 2019: (Start)

%F a(n) = a(n-3) + 5 with a(1) = 1, a(2) = 2, a(3) = 4 for n > 3.

%F a(n) = (15*n - 9 + (w^(2*n) - w^n)*(1 + 2*w))/9 where w = (-1 + sqrt(-3))/2. (End)

%F 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

%p 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

%t Select[Range[0, 200], MemberQ[{1, 2, 4}, Mod[#, 5]] &] (* _Vladimir Joseph Stephan Orlovsky_, Feb 11 2012 *)

%t LinearRecurrence[{1,0,1,-1},{1,2,4,6},90] (* _Harvey P. Dale_, May 20 2019 *)

%o (Magma)[ n: n in [0..120] | n mod 5 in {1, 2, 4} ]; // _Vincenzo Librandi_, Dec 29 2010

%o (PARI) a(n)=n\3*5+[-1,1,2][n%3+1] \\ _Charles R Greathouse IV_, Jan 18 2012

%o (Sage) [(15*n - 9 + 2*sqrt(3)*sin(2*n*pi/3))/9 for n in (1..100)] # _G. C. Greubel_, Nov 06 2019

%o (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

%Y Cf. A001622, A032794, A032795.

%K nonn,easy

%O 1,2

%A _Patrick De Geest_, May 15 1998

%E Better description from _Michael Somos_, Jun 08 2000