A131737 Essentially even numbers followed by duplicated odd numbers.

0, 1, 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45

Offset: 0

Author

Paul Curtz, Sep 19 2007

Keywords

• nonn
• easy
• less

Examples

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

Links

• Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1).

Crossrefs

Cf. A004396.

Programs

Maple

A131737 := proc(n): (1/9)*add(5*(k mod 3)+2*((k+1) mod 3)-((k+2) mod 3),k=0..n)-1+(binomial(2*n,n) mod 2)+(binomial((n+1)^2,n+3) mod 2) end: seq( A131737(n),n=0..74); # Johannes W. Meijer, Jun 27 2011

Mathematica

Join[{0, 1}, LinearRecurrence[{1, 0, 1, -1}, {1, 1, 2, 3}, 68]] (* Georg Fischer, Feb 27 2019 *)
Insert[Flatten[Table[If[EvenQ[n],n,{n,n}],{n,0,70}]],1,2] (* Harvey P. Dale, Sep 04 2020 *)

PARI

{a(n) = (n==0) + (n==1) + (n\3)*2 + (n%3) - 1}; /* Michael Somos, Jan 11 2011 */

Formula

a(0)=0. a(1)=a(2)=1. a(3*n)=A005408(n-1). a(3*n+1)=a(3*n)+1. a(3*n+2)=a(3*n)+2, n>0.

O.g.f.: x*(1+x^4)/((1-x)^2*(x^2+x+1)). a(n)=(2*n-2-A057078(n))/3, n>1. - R. J. Mathar, Jul 16 2008

Euler transform of length 8 sequence [ 1, 0, 1, 1, 0, 0, 0, -1]. - Michael Somos, Jan 11 2011

0 = a(n) - a(n+1) - a(n+3) + a(n+4) if n>1. - Michael Somos, Nov 11 2015

a(n) = floor((2*n-1)/3) for n > 1. - Werner Schulte, Feb 27 2019

Extensions

Edited by R. J. Mathar, Jul 16 2008