A130823 Each odd number appears thrice.

1, 1, 1, 3, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 9, 11, 11, 11, 13, 13, 13, 15, 15, 15, 17, 17, 17, 19, 19, 19, 21, 21, 21, 23, 23, 23, 25, 25, 25, 27, 27, 27, 29, 29, 29, 31, 31, 31, 33, 33, 33, 35, 35, 35, 37, 37, 37, 39, 39, 39, 41, 41, 41, 43, 43, 43, 45, 45, 45, 47, 47, 47, 49, 49

Offset: 1

Author

Paul Curtz, Jul 17 2007

Keywords

• nonn
• easy

Comments

Partial sums of 1,0,0,2,0,0,2,0,0,2,0,0,... . - Emeric Deutsch, Jul 23 2007

Examples

			G.f. = x + x^2 + x^3 + 3*x^4 + 3*x^5 + 3*x^6 + 5*x^7 + 5*x^8 + 5*x^9 + 7*x^10 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Programs

Magma

[Floor((n-1)/3)*2+1: n in [1..80]]; // Vincenzo Librandi, Aug 10 2011

Maple

G:=x*(1+x^3)/((1-x)*(1-x^3)): Gser:=series(G,x=0,82): seq(coeff(Gser,x,n),n= 1..75); # Emeric Deutsch, Jul 23 2007

Mathematica

Flatten[Table[n,{n,1,49,2},{3}]] (* or *) LinearRecurrence[{1,0,1,-1},{1,1,1,3},100] (* or *) Accumulate[PadRight[{1},100,{2,0,0}]] (* Harvey P. Dale, Apr 20 2015 *)

PARI

{a(n) = (n-1)\3*2+1}; \\ Michael Somos, Aug 16 2007

Formula

a(n) = 2*floor((n-1)/3) + 1. - Joerg Arndt, Jan 02 2024 [corrected by Aadesh Tikhe, 06 May 2025]

G.f.: x*(1 + x^3)/((1 - x)*(1 - x^3)). - Emeric Deutsch, Jul 23 2007

From Michael Somos, Aug 16 2007: (Start)

Euler transform of length 6 sequence [1, 0, 2, 0, 0, -1].

a(n + 3) = a(n) + 2.

a(n) = - a(1-n) for all n in Z. (End)

a(n) = floor((n-1)*(n+1)/3) - floor((n-2)*n/3). - Bruno Berselli, Mar 03 2017

a(n) = (6*n-3-4*sqrt(3)*sin(2*(n-2)*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017

Extensions

More terms from Emeric Deutsch, Jul 23 2007