A098180 Odd numbers with twice the odd numbers repeated in order between them.

1, 2, 2, 3, 5, 6, 6, 7, 9, 10, 10, 11, 13, 14, 14, 15, 17, 18, 18, 19, 21, 22, 22, 23, 25, 26, 26, 27, 29, 30, 30, 31, 33, 34, 34, 35, 37, 38, 38, 39, 41, 42, 42, 43, 45, 46, 46, 47, 49, 50, 50, 51, 53, 54, 54, 55, 57, 58, 58, 59, 61, 62, 62, 63, 65, 66, 66, 67, 69, 70, 70, 71

Offset

0,2

Comments

Partial sums of A098178.

Also A042968 with the even terms repeated. - Michel Marcus, Apr 14 2015

Fixed points are [2,3,6,7,10,11,..] = A042964. - Wesley Ivan Hurt, Oct 13 2015

Links

Iain Fox, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Formula

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

a(n) = sqrt(2)*sin(Pi*n/2+Pi/4)/2+n+1/2.

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

From Wesley Ivan Hurt, Apr 12 2015, Oct 13 2015: (Start)

a(n) = (2n+1-(-1)^((n+1)*(n+2)/2))/2.

a(n) = n + A002265(n) - A002265(n-2). (End)

E.g.f: (exp(-i*x)*((1+i) + (1-i)*exp(2*i*x) + exp((1+i)*x)*(2+4*x)))/4, where i = sqrt(-1). - Iain Fox, Oct 17 2018

Maple

A098180:=n->(2*n+1-(-1)^((n+1)*(n+2)/2))/2: seq(A098180(n), n=0..100); # Wesley Ivan Hurt, Apr 12 2015

Mathematica

Table[(2 n + 1 - (-1)^((n + 1) (n + 2)/2))/2, {n, 0, 40}] (* Wesley Ivan Hurt, Apr 12 2015 *)

Prog

(Magma) [Floor((2*n+1-(-1)^((n+1)*(n+2)/2))/2): n in [0..80]]; // Vincenzo Librandi, Apr 13 2015
(PARI) first(n) = Vec((1+x)*(1-x+x^2)/((1-x)^2*(1+x^2)) + O(x^n)) \\ Iain Fox, Oct 17 2018
(PARI) a(n) = (2*n+1-(-1)^((n+1)*(n+2)/2))/2 \\ Iain Fox, Oct 17 2018

Crossrefs

Cf. A002265, A042964, A042968, A098178.

Sequence in context: A308773 A051732 A098382 * A357411 A270875 A117752

Adjacent sequences: A098177 A098178 A098179 * A098181 A098182 A098183

Keyword

easy,nonn

Author

Paul Barry, Aug 30 2004

Status

approved