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A098181 Odd numbers with two times the positive even numbers repeated in order between them. 4
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

Table of n, a(n) for n=0..71.

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) = (2n+3)/2-cos(Pi*n/2)/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(4n) = 4n+1, a(4n+1) = 4n+3, a(4n+2) = 4n+4, a(4n+3) = 4n+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

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 + ...

MATHEMATICA

Table[Floor[Binomial[n + 3, 2]/2] - Floor[Binomial[n + 1, 2]/2], {n,

  0, 71}] (* or *)

CoefficientList[Series[(1 + x)/((1 - x)^2 (1 + x^2)), {x, 0, 71}], 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 */

CROSSREFS

Cf. A098180.

Sequence in context: A113454 A157726 A082223 * A111914 A051665 A028263

Adjacent sequences:  A098178 A098179 A098180 * A098182 A098183 A098184

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Aug 30 2004

STATUS

approved

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Last modified December 5 21:40 EST 2016. Contains 278771 sequences.