A123920 - OEIS

%I #21 Sep 17 2024 08:24:33

%S 1,2,1,2,2,2,3,4,3,4,4,4,5,6,5,6,6,6,7,8,7,8,8,8,9,10,9,10,10,10,11,

%T 12,11,12,12,12,13,14,13,14,14,14,15,16,15,16,16,16,17,18,17,18,18,18,

%U 19,20,19,20,20,20,21,22,21,22,22,22,23,24,23,24,24,24,25,26,25,26,26,26

%N Number of numbers congruent to 2 or 4 mod 6 between n and 2n inclusive.

%H G. C. Greubel, <a href="/A123920/b123920.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,1,-1).

%F a(n) = 2k - 1 for n = {6k - 5, 6k - 3}, where k = 1,2,3,... a(n) = 2k for n = {6k - 4, 6k - 2, 6k - 1, 6k}, where k = 1,2,3,... - _Alexander Adamchuk_, Nov 08 2006

%F G.f.: x*(1+x-x^2+x^3)/((1-x)*(1-x^6)). - _G. C. Greubel_, Aug 07 2019

%p seq(coeff(series(x*(1+x-x^2+x^3)/((1-x)*(1-x^6)), x, n+1), x, n), n = 1..80); # _G. C. Greubel_, Aug 07 2019

%t f[n_]:= Floor[n/2] - Floor[n/6]; Table[f[2n] - f[n-1], {n, 80}] (* _Robert G. Wilson v_ *)

%t Table[Count[Range[n,2n],_?(MemberQ[{2,4},Mod[#,6]]&)],{n,80}] (* _Harvey P. Dale_, Mar 25 2019 *)

%t LinearRecurrence[{1,0,0,0,0,1,-1}, {1,2,1,2,2,2,3}, 80] (* _G. C. Greubel_, Aug 07 2019 *)

%o (PARI) my(x='x+O('x^80)); Vec(x*(1+x-x^2+x^3)/((1-x)*(1-x^6))) \\ _G. C. Greubel_, Aug 07 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( x*(1+x-x^2+x^3)/((1-x)*(1-x^6)) )); // _G. C. Greubel_, Aug 07 2019

%o (Sage)

%o def A123920_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( x*(1+x-x^2+x^3)/((1-x)*(1-x^6)) ).list()

%o a=A123920_list(80); a[1:] # _G. C. Greubel_, Aug 07 2019

%o (GAP) a:=[1,2,1,2,2,2,3];; for n in [8..80] do a[n]:=a[n-1]+a[n-6]-a[n-7]; od; a; # _G. C. Greubel_, Aug 07 2019

%Y Cf. A123919.

%K nonn,easy

%O 1,2

%A _Giovanni Teofilatto_, Oct 29 2006

%E Corrected and extended by _Robert G. Wilson v_, Oct 29 2006

%E More terms from _Alexander Adamchuk_, Nov 08 2006