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A098181 Two consecutive odd numbers separated by multiples of four, repeated twice, between them, written in increasing order. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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).

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

EXAMPLE

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 + ...

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 *)

PROG

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

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

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

(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

CROSSREFS

Cf. A098180.

Sequence in context: A157726 A082223 A292351 * A322407 A111914 A051665

Adjacent sequences: A098178 A098179 A098180 * A098182 A098183 A098184

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Aug 30 2004

EXTENSIONS

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

STATUS

approved

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Last modified December 5 17:16 EST 2022. Contains 358588 sequences. (Running on oeis4.)