A098181 Two consecutive odd numbers separated by multiples of four, repeated twice, between them, written in increasing order.

1, 3, 4, 4, 5, 7, 8, 8, 9, 11, 12, 12, 13, 15, 16, 16, 17, 19, 20, 20, 21, 23, 24, 24, 25, 27, 28, 28, 29, 31, 32, 32, 33, 35, 36, 36, 37, 39, 40, 40, 41, 43, 44, 44, 45, 47, 48, 48, 49, 51, 52, 52, 53, 55, 56, 56, 57, 59, 60, 60, 61, 63, 64, 64, 65, 67, 68, 68, 69, 71, 72, 72

Offset: 0

Author

Paul Barry, Aug 30 2004

Keywords

• easy
• nonn

Comments

Essentially partial sums of A007877.

a(n) is the number of odd coefficients of the q-binomial coefficient [n+2 choose 2]. (Easy to prove.) - Richard Stanley, Oct 12 2016

Examples

			G.f. = 1 + 3*x + 4*x^2 + 4*x^3 + 5*x^4 + 7*x^5 + 8*x^6 + 8*x^7 + 9*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• P. Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A098180.

Programs

GAP

a:=[1,3,4,4];; for n in [5..80] do a[n]:=2*a[n-1]-2*a[n-2]+2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, May 22 2019

Magma

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( (1+x)/((1-x)^2*(1+x^2)) )); // G. C. Greubel, May 22 2019

Maple

A:=seq((2*n+3 - cos(Pi*n/2) + sin(Pi*n/2))/2, n=0..50); \\ Bernard Schott, Jun 07 2019

Mathematica

Table[Floor[Binomial[n+3, 2]/2] -Floor[Binomial[n+1, 2]/2], {n, 0, 80}] (* or *) CoefficientList[Series[(1+x)/((1-x)^2*(1+x^2)), {x, 0, 80}], x] (* Michael De Vlieger, Oct 12 2016 *)

PARI

{a(n) = n\4*4 + [1, 3, 4, 4][n%4+1]}; /* Michael Somos, Sep 11 2014 */

Sage

((1+x)/((1-x)^2*(1+x^2))).series(x, 80).coefficients(x, sparse=False) # G. C. Greubel, May 22 2019

Formula

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

a(n) = ( (2*n+3) - cos(Pi*n/2) + sin(Pi*n/2) )/2.

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4).

a(n) = floor(C(n+3, 2)/2)-floor(C(n+1, 2)/2). - Paul Barry, Jan 01 2005

a(4*n) = 4*n+1, a(4*n+1) = 4*n+3, a(4*n+2) = a(4*n+3) = 4*n+4. - Philippe Deléham, Apr 06 2007

Euler transform of length 4 sequence [ 3, -2, 0, 1]. - Michael Somos, Sep 11 2014

a(-3-n) = -a(n) for all n in Z. - Michael Somos, Sep 11 2014

a(n) = log_2(|A174882(n+2)|). [Barry] - R. J. Mathar, Aug 18 2017

a(n) = (2*n+3 - (-1)^ceiling(n/2))/2. - Wesley Ivan Hurt, Sep 29 2017

Extensions

Name edited by G. C. Greubel, Jun 06 2019