

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
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OFFSET

0,2


COMMENTS

Essentially partial sums of A007877.
a(n) is the number of odd coefficients of the qbinomial 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)/((1x)^2(1+x^2)).
a(n) = (2n+3)/2cos(Pi*n/2)/2+sin(Pi*n/2)/2.
a(n) = 2*a(n1)2*a(n2)+2*a(n3)a(n4).
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(3n) = 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



