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A047220 Numbers that are congruent to {0, 1, 3} mod 5. 24
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, 41, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 75, 76, 78, 80, 81, 83, 85, 86, 88, 90, 91, 93, 95, 96, 98, 100, 101, 103, 105, 106 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
First differences are (1,2,2), repeat, with period 3 (A130196). - N. J. A. Sloane, Dec 03 2015
Also numbers k such that k*(k+2)*(k+4) is divisible by 5. - Bruno Berselli, Dec 28 2017
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
LINKS
Allan Bickle, Nordhaus-Gaddum Theorems for k-Decompositions, Congr. Num. 211 (2012) 171-183.
Z. Furedi, A. Kostochka, M. Stiebitz, R. Skrekovski, and D. West, Nordhaus-Gaddum-type theorems for decompositions into many parts, J. Graph Theory 50 (2005), 273-292.
FORMULA
a(n) = floor(5*(n-1)/3). - Gary Detlefs, Feb 20 2010
a(n) = 2*n - floor(n/3) - (n^2 mod 3), with offset 0. - Gary Detlefs, Mar 19 2010
G.f.: x^2*(1 + 2*x + 2*x^2)/(1 - x)^2/(1 + x + x^2). - Colin Barker, Feb 17 2012
a(n) = n + floor(2*(n-1)/3) - 1. - Arkadiusz Wesolowski, Sep 18 2012
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 5*n/3 - 2 + 2*sin(2*n*Pi/3)/(3*sqrt(3)).
a(3*k) = 5*k-2, a(3*k-1) = 5*k-4, a(3*k-2) = 5*k-5. (End)
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
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
MAPLE
seq(floor(5*(n-1)/3), n=1..56); # Gary Detlefs, Feb 20 2010
seq(2*n-floor(n/3)-(n^2 mod 3), n=0..55); # Gary Detlefs, Mar 19 2010
MATHEMATICA
Table[Floor[5*(n-1)/3], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
PROG
(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
(PARI) a(n)=n + 2*(n-1)\3 - 1 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
Cf. A001622, A011655, A130196 (first differences).
Sequence in context: A247913 A188046 A244644 * A329845 A329993 A064994
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified February 20 19:28 EST 2024. Contains 370217 sequences. (Running on oeis4.)