A123919 Number of numbers congruent to 2 or 4 mod 6 and <= n.

0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 26, 26, 26

Offset: 1

Author

Giovanni Teofilatto, Oct 29 2006

Keywords

• easy
• nonn

Comments

First differences of A056827. - R. J. Mathar, Nov 22 2008

a(n+2) is the graph radius of the n X n knight graph for n > 7. - Eric W. Weisstein, Nov 20 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Graph Radius
• Eric Weisstein's World of Mathematics, Knight Graph
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Crossrefs

Cf. A047235, A056827, A120325, A004526, A152467.

Programs

GAP

a:=[0,1,1,2,2,2,2];; for n in [8..80] do a[n]:=a[n-1]+a[n-6]-a[n-7]; od; a; # G. C. Greubel, Aug 07 2019

Magma

[Floor(n/2) - Floor(n/6) : n in [1..100]]; // Wesley Ivan Hurt, Apr 26 2021

Mathematica

a[n_] := Floor[n/2] - Floor[n/6]; Array[a, 80] (* Robert G. Wilson v, Oct 29 2006 *)
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 1, 2, 2, 2, 2}, 80] (* G. C. Greubel, Aug 07 2019 *)

PARI

my(x='x+O('x^80)); concat([0], Vec(x^2*(1+x^2)/((1-x)*(1-x^6)))) \\ G. C. Greubel, Aug 07 2019

PARI

a(n) = floor(n/2) - floor(n/6);  \\ Joerg Arndt, Nov 23 2019

Formula

a(n) = floor(n/2) - floor(n/6).

From R. J. Mathar, Nov 22 2008: (Start)

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

a(n+1) - a(n) = A120325(n+1). (End)

a(n) = A004526(n) - A152467(n). - Omar E. Pol, Nov 25 2019

a(n) = a(n-1)+a(n-6)-a(n-7). - Wesley Ivan Hurt, Apr 26 2021

a(n) = floor((2*n+3+(-1)^n)/6). - Adriano Caroli, Mar 14 2025