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

0,5

Links

Table of n, a(n) for n=0..68.

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

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Cf. A004396.

Sequence in context: A317686 A156689 A168052 * A004396 A066481 A248103

Adjacent sequences: A131734 A131735 A131736 * A131738 A131739 A131740

Keyword

nonn,easy,less

Author

Paul Curtz, Sep 19 2007

Extensions

Edited by R. J. Mathar, Jul 16 2008

Status

approved