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%I #53 May 03 2023 15:30:20
%S 1,2,3,4,7,8,9,10,13,14,15,16,19,20,21,22,25,26,27,28,31,32,33,34,37,
%T 38,39,40,43,44,45,46,49,50,51,52,55,56,57,58,61,62,63,64,67,68,69,70,
%U 73,74,75,76,79,80,81,82,85,86,87,88,91,92,93,94,97,98
%N Numbers that are congruent to {1, 2, 3, 4} mod 6.
%C a(k)^m is a term for k and m in N. - _Jerzy R Borysowicz_, Apr 18 2023
%H Vincenzo Librandi, <a href="/A047227/b047227.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F From _Johannes W. Meijer_, Jul 07 2011: (Start)
%F a(n) = floor((n+2)/4) + floor((n+1)/4) + floor(n/4) + 2*floor((n-1)/4) + floor((n+3)/4).
%F G.f.: x*(1 + x + x^2 + x^3 + 2*x^4)/(x^5 - x^4 - x + 1). (End)
%F From _Wesley Ivan Hurt_, May 20 2016: (Start)
%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F a(n) = (6n - 5 - i^(2n) + (1+i)*i^(1-n) + (1-i)*i^(n-1))/4 where i=sqrt(-1).
%F a(2n) = A047235(n), a(2n-1) = A047241(n). (End)
%F E.g.f.: (4 + sin(x) - cos(x) + (3*x - 2)*sinh(x) + 3*(x - 1)*cosh(x))/2. - _Ilya Gutkovskiy_, May 21 2016
%F From _Wesley Ivan Hurt_, May 21 2016: (Start)
%F a(n) = A047246(n) + 1.
%F a(n+2) - a(n+1) = A093148(n) for n>0.
%F a(1-n) = - A047247(n). (End)
%F Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 + 2*log(2)/3 - log(3)/4. - _Amiram Eldar_, Dec 17 2021
%p A047227:=n->(6*n-5-I^(2*n)+(1+I)*I^(1-n)+(1-I)*I^(n-1))/4: seq(A047227(n), n=1..100); # _Wesley Ivan Hurt_, May 20 2016
%t Complement[Range[100], Flatten[Table[{6n - 1, 6n}, {n, 0, 15}]]] (* _Alonso del Arte_, Jul 07 2011 *)
%t Select[Range[100], MemberQ[{1, 2, 3, 4}, Mod[#, 6]]&] (* _Vincenzo Librandi_, Jan 06 2013 *)
%o (Magma) [n: n in [0..100] | n mod 6 in [1..4]]; // _Vincenzo Librandi_, Jan 06 2013
%o (PARI) a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,0,1]^(n-1)*[1;2;3;4;7])[1,1] \\ _Charles R Greathouse IV_, May 03 2023
%Y Cf. A008621, A047235, A047241, A047245, A047246, A047247, A093148.
%Y Complement of A047264. Equals A203016 divided by 3.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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