A047261 - OEIS

%I #47 Jul 27 2024 12:02:25

%S 2,4,5,8,10,11,14,16,17,20,22,23,26,28,29,32,34,35,38,40,41,44,46,47,

%T 50,52,53,56,58,59,62,64,65,68,70,71,74,76,77,80,82,83,86,88,89,92,94,

%U 95,98,100,101,104,106,107,110,112,113,116,118,119,122,124

%N Numbers that are congruent to {2, 4, 5} mod 6.

%C If B and C are terms in the sequence then 2*B*C is a term. B (resp. C) is a term iff B (resp. C) mod 6 = 2, 4 or 5. It follows that (2*B*C) mod 6 = (2*(B mod 6)*(C mod 6)) mod 6 = 2 or 4 and therefore 2*B*C is a term. Examples: for B=16 and C=29, 2*16*29 = 928 is a term: (2*B*C) mod 6 = (2*16*29) mod 6 = 4; (2*2*2) mod 6 = 2. - _Jerzy R Borysowicz_, May 24 2018

%H Reinhard Zumkeller, <a href="/A047261/b047261.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 (1,0,1,-1).

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

%F A214090(a(n)) = 1. - _Reinhard Zumkeller_, Jul 06 2012

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

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

%F a(n) = (6*n - 1 - 2*cos(2*n*Pi/3))/3.

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

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 - log(2+sqrt(3))/(2*sqrt(3)) + log(2)/3. - _Amiram Eldar_, Dec 16 2021

%F E.g.f.: (3 + exp(x)*(6*x - 1) - 2*exp(-x/2)*cos(sqrt(3)*x/2))/3. - _Stefano Spezia_, Jul 26 2024

%p A047261:=n->(6*n-1-2*cos(2*n*Pi/3))/3: seq(A047261(n), n=1..100); # _Wesley Ivan Hurt_, Jun 14 2016

%t CoefficientList[Series[(1 + x)*(x^2 + 2)/((1 + x + x^2)*(x - 1)^2), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Aug 16 2014 *)

%t Select[ Range@ 125, MemberQ[{2, 4, 5}, Mod[#, 6]] &] (* or *)

%t LinearRecurrence[{1, 0, 1, -1}, {2, 4, 5, 8}, 62] (* _Robert G. Wilson v_, Jun 13 2018 *)

%o (Haskell)

%o a047261 n = a047261_list !! n

%o a047261_list = 2 : 4 : 5 : map (+ 6) a047261_list

%o -- _Reinhard Zumkeller_, Feb 19 2013, Jul 06 2012

%o (Magma) [n : n in [0..150] | n mod 6 in [2, 4, 5]]; // _Wesley Ivan Hurt_, Jun 14 2016

%Y Cf. A047242 (complement).

%Y Cf. A007310, A047228, A047241, A047273, A056970, A214090.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Wesley Ivan Hurt_, Aug 16 2014