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

Author

Paul Barry, Aug 30 2004

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A002265, A042964, A042968, A098178.

Programs

Magma

[Floor((2*n+1-(-1)^((n+1)*(n+2)/2))/2): n in [0..80]]; // Vincenzo Librandi, Apr 13 2015

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 *)

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

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