A047245 - OEIS

%I #56 Aug 01 2023 19:41:39

%S 1,2,3,7,8,9,13,14,15,19,20,21,25,26,27,31,32,33,37,38,39,43,44,45,49,

%T 50,51,55,56,57,61,62,63,67,68,69,73,74,75,79,80,81,85,86,87,91,92,93,

%U 97,98,99,103,104,105,109,110,111,115,116,117,121,122,123

%N Numbers that are congruent to {1, 2, 3} mod 6.

%C a(k)^m is a term iff {a(k) is odd and m is a nonnegative integer} or {m is in A004273}. - _Jerzy R Borysowicz_, May 08 2023

%H David Lovler, <a href="/A047245/b047245.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 From _Johannes W. Meijer_, Jun 07 2011: (Start)

%F a(n) = ceiling(n/3) + ceiling((n-1)/3) + ceiling((n-2)/3) + 3*ceiling((n-3)/3).

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

%F a(n) = 3*floor((n-1)/3) + n. - _Gary Detlefs_, Dec 22 2011

%F From _Wesley Ivan Hurt_, Apr 13 2015: (Start)

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

%F a(n) = 2*n-3 + ((2*n-3) mod 3). (End)

%F From _Wesley Ivan Hurt_, Jun 13 2016: (Start)

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

%F a(3k) = 6k-3, a(3k-1) = 6k-4, a(3k-2) = 6k-5. (End)

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (9-2*sqrt(3))*Pi/36 + log(2+sqrt(3))/(2*sqrt(3)) - log(2)/6. - _Amiram Eldar_, Dec 14 2021

%p A047245:=n->2*n-3+((2*n-3) mod 3): seq(A047245(n), n=1..100); # _Wesley Ivan Hurt_, Apr 13 2015

%t Select[Range[0, 200], Mod[#, 6] == 1 || Mod[#, 6] == 2 || Mod[#, 6] == 3 &] (* _Vladimir Joseph Stephan Orlovsky_, Jul 07 2011 *)

%t Flatten[Table[{6n + 1, 6n + 2, 6n + 3}, {n, 0, 19}]] (* _Alonso del Arte_, Jul 07 2011 *)

%t Select[Range[0, 200], MemberQ[{1, 2, 3}, Mod[#, 6]] &] (* _Vincenzo Librandi_, Apr 14 2015 *)

%o (Magma) [2*n-3+((2*n-3) mod 3) : n in [1..100]]; // _Wesley Ivan Hurt_, Apr 13 2015

%o (PARI) a(n) = 3*floor((n-1)/3) + n; \\ _David Lovler_, Aug 03 2022

%Y Cf. A047240, A047244, A047258 (complement).

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_