A047218 Numbers that are congruent to {0, 3} mod 5.

0, 3, 5, 8, 10, 13, 15, 18, 20, 23, 25, 28, 30, 33, 35, 38, 40, 43, 45, 48, 50, 53, 55, 58, 60, 63, 65, 68, 70, 73, 75, 78, 80, 83, 85, 88, 90, 93, 95, 98, 100, 103, 105, 108, 110, 113, 115, 118, 120, 123, 125, 128, 130, 133, 135, 138, 140, 143, 145, 148

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Multiples of 5 interleaved with 2 less than multiples of 5. - Wesley Ivan Hurt, Oct 19 2013

Numbers k such that k^2/5 + k*(k + 1)/10 = k*(3*k + 1)/10 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..10001
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A001622, A047211, A047212, A047216.

Programs

Maple

seq(floor(5*k/2)-2, k=1..100); # Wesley Ivan Hurt, Sep 27 2013

Mathematica

Select[Range[0, 200], MemberQ[{0, 3}, Mod[#, 5]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)
Table[Floor[5 n/2] - 2, {n,100}] (* Wesley Ivan Hurt, Sep 27 2013 *)
With[{c5=5*Range[0,30]},Riffle[c5,c5+3]] (* or *) LinearRecurrence[{1,1,-1},{0,3,5},60] (* Harvey P. Dale, Apr 02 2017 *)

PARI

forstep(n=0,200,[3,2],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

Formula

a(n) = 2*n - 5 + ceiling(n/2). - Jesus De Loera (deloera(AT)math.ucdavis.edu)

a(n) = 5*n - a(n-1) - 7 for n>1, a(1)=0. - Vincenzo Librandi, Aug 05 2010

From Bruno Berselli, Jun 28 2011: (Start)

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

a(n) = (10*n + (-1)^n - 9)/4.

a(n) = a(n-1) + a(n-2) - a(n-3). (End)

a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=3 and b(k)=A020714(k-1)=5*2^(k-1) for k>0. - Philippe Deléham, Oct 17 2011

a(n) = n + ceiling(3*(n-1)/2) - 1. - Arkadiusz Wesolowski, Sep 18 2012

a(n) = floor(5*n/2)-2 = 3*n - 3 - floor((n-1)/2). - Wesley Ivan Hurt, Oct 14 2013

a(n+1) = n + (n + (n + (n mod 2))/2). - Wesley Ivan Hurt, Oct 19 2013

Sum_{n>=2} (-1)^n/a(n) = log(5)/4 - sqrt(5)*log(phi)/10 - sqrt(1-2/sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021

E.g.f.: 2 + ((5*x - 9/2)*exp(x) + (1/2)*exp(-x))/2. - David Lovler, Aug 22 2022