A047220 - OEIS

%I #64 Aug 15 2024 10:58:06

%S 0,1,3,5,6,8,10,11,13,15,16,18,20,21,23,25,26,28,30,31,33,35,36,38,40,

%T 41,43,45,46,48,50,51,53,55,56,58,60,61,63,65,66,68,70,71,73,75,76,78,

%U 80,81,83,85,86,88,90,91,93,95,96,98,100,101,103,105,106

%N Numbers that are congruent to {0, 1, 3} mod 5.

%C First differences are (1,2,2), repeat, with period 3 (A130196). - _N. J. A. Sloane_, Dec 03 2015

%C Also numbers k such that k*(k+2)*(k+4) is divisible by 5. - _Bruno Berselli_, Dec 28 2017

%C Maximum sum of degeneracies over all decompositions of the complete graph of order n into four factors. The extremal decompositions are characterized in the Bickle link below. - _Allan Bickle_, Dec 21 2021

%H Vincenzo Librandi, <a href="/A047220/b047220.txt">Table of n, a(n) for n = 1..1000</a>

%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/ngcorepaper.pdf">Nordhaus-Gaddum Theorems for k-Decompositions</a>, Congr. Num. 211 (2012) 171-183.

%H Z. Füredi, A. Kostochka, M. Stiebitz, R. Skrekovski, and D. West, <a href="https://faculty.math.illinois.edu/~west/pubs/norgad.pdf">Nordhaus-Gaddum-type theorems for decompositions into many parts</a>, J. Graph Theory 50 (2005), 273-292.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

%F a(n) = floor(5*(n-1)/3). - _Gary Detlefs_, Feb 20 2010

%F a(n) = 2*n - floor(n/3) - (n^2 mod 3), with offset 0. - _Gary Detlefs_, Mar 19 2010

%F G.f.: x^2*(1 + 2*x + 2*x^2)/(1 - x)^2/(1 + x + x^2). - _Colin Barker_, Feb 17 2012

%F a(n) = n + floor(2*(n-1)/3) - 1. - _Arkadiusz Wesolowski_, Sep 18 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) = 5*n/3 - 2 + 2*sin(2*n*Pi/3)/(3*sqrt(3)).

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

%F E.g.f.: 2 + (5*x - 6)*exp(x)/3 + 2*sin(sqrt(3)*x/2)*(cosh(x/2) - sinh(x/2))/(3*sqrt(3)). - _Ilya Gutkovskiy_, Jun 14 2016

%F Sum_{n>=2} (-1)^n/a(n) = sqrt(1-2/sqrt(5))*Pi/5 + 2*log(phi)/sqrt(5) + log(2)/5, where phi is the golden ratio (A001622). - _Amiram Eldar_, Apr 16 2023

%p seq(floor(5*(n-1)/3), n=1..56); # _Gary Detlefs_, Feb 20 2010

%p seq(2*n-floor(n/3)-(n^2 mod 3), n=0..55); # _Gary Detlefs_, Mar 19 2010

%t Table[Floor[5*(n-1)/3], {n,100}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 28 2012 *)

%o (Magma) I:=[0, 1, 3, 5]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // _Vincenzo Librandi_, Apr 26 2012

%o (PARI) a(n)=n + 2*(n-1)\3 - 1 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Cf. A001622, A011655, A130196 (first differences).

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_