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Numbers that are congruent to {1, 3, 4, 5} (mod 6).
4

%I #41 Feb 27 2024 14:41:29

%S 1,3,4,5,7,9,10,11,13,15,16,17,19,21,22,23,25,27,28,29,31,33,34,35,37,

%T 39,40,41,43,45,46,47,49,51,52,53,55,57,58,59,61,63,64,65,67,69,70,71,

%U 73,75,76,77,79,81,82,83,85,87,88,89,91,93,94,95,97,99

%N Numbers that are congruent to {1, 3, 4, 5} (mod 6).

%C "Polyrhythmic Sequence" P(2,3): numbers congruent to 1 (mod 2) and 1 (mod 3). (See A267027 for definition and description.) - _Bob Selcoe_, Jan 12 2016

%H Colin Barker, <a href="/A047251/b047251.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).

%F From _R. J. Mathar_, Oct 08 2011: (Start)

%F a(n) = 3*n/2 - 1/2 - cos(Pi*n/2)/2.

%F G.f.: x*(x^3+x+1)/((x-1)^2*(x^2+1)). (End)

%F a(n) = (-2 - (-i)^n - i^n + 6n)/4, with i=sqrt(-1). - _Colin Barker_, Oct 19 2015

%F From _Wesley Ivan Hurt_, May 31 2016: (Start)

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

%F a(2k) = A047270(k), a(2k-1) = A016777(k-1) for n>0. (End)

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 5*sqrt(3)*Pi/36 - log(2)/3 + log(3)/4. - _Amiram Eldar_, Dec 17 2021

%p A047251:=n->(-2-(-I)^n-I^n+6*n)/4: seq(A047251(n), n=1..100); # _Wesley Ivan Hurt_, May 31 2016

%t Select[Range[0, 200], MemberQ[{1, 3, 4, 5}, Mod[#, 6]] &] (* _Vincenzo Librandi_, Jan 12 2016 *)

%t LinearRecurrence[{2,-2,2,-1},{1,3,4,5},70] (* _Harvey P. Dale_, Feb 27 2024 *)

%o (PARI) a(n) = (-2-(-I)^n-I^n+6*n)/4 \\ _Colin Barker_, Oct 19 2015

%o (PARI) Vec(x*(x^3+x+1)/((x-1)^2*(x^2+1)) + O(x^100)) \\ _Colin Barker_, Oct 19 2015

%o (Magma) [n: n in [0..150]|n mod 6 in {1,3,4,5}]; // _Vincenzo Librandi_, Jan 12 2016

%Y Cf. A016777, A047270, A267027.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_