A029579 a(2*n) = n+1, a(2*n-1) = 2*n-1.

1, 1, 2, 3, 3, 5, 4, 7, 5, 9, 6, 11, 7, 13, 8, 15, 9, 17, 10, 19, 11, 21, 12, 23, 13, 25, 14, 27, 15, 29, 16, 31, 17, 33, 18, 35, 19, 37, 20, 39, 21, 41, 22, 43, 23, 45, 24, 47, 25, 49, 26, 51, 27, 53, 28, 55, 29, 57, 30, 59, 31, 61, 32, 63, 33, 65, 34, 67, 35, 69, 36, 71, 37

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A026741, A174239.

Programs

GAP

a:=[1,1,2,3];; for n in [5..80] do a[n]:=2*a[n-2]-a[n-4]; od; a; # Muniru A Asiru, Sep 21 2018

Haskell

import Data.List (transpose)
a029579 n = if m == 0 then n' + 1 else n  where (n', m) = divMod n 2
a029579_list = concat $ transpose [[1 ..], [1, 3 ..]]
-- Reinhard Zumkeller, Apr 06 2015

Maple

seq(coeff(series((1+x+x^3)/(1-x^2)^2,x,n+1), x, n), n = 0 .. 80); # Muniru A Asiru, Sep 21 2018

Mathematica

With[{nn = 50}, Riffle[Range[nn], Range[1, 2 nn + 1, 2]]] (* or *) LinearRecurrence[{0, 2, 0, -1}, {1, 1, 2, 3}, 120] (* Harvey P. Dale, Apr 22 2018 *)

PARI

a(n)=(3*n+2+(2-n)*(-1)^n)/4 \\ Charles R Greathouse IV, Sep 02 2015

Formula

G.f.: (1 + x + x^3)/(1 - x^2)^2.

From Paul Barry, Aug 29 2004: (Start)

a(n) = (3*n + 2)/4 + (2 - n)*(-1)^n/4.

a(n) = 2*a(n-2) - a(n-4).

Binomial transform is A098156. (End)

From Gary W. Adamson, May 08 2010: (Start)

Let M = an infinite lower triangular matrix with (1, 1, 0, 1, 0, 0, 0, ...) in every column; for columns > 0, shifted down twice from the previous column.

Then A029579 = M * [1, 2, 3, 0, 0, 0, ...]. (End)

From Paul Curtz, Sep 21 2018: (Start)

a(-n) = -A174239(n).

Terms of A026741(n+1) swapped in pairs. (End)

Extensions

Title simplified by Sean A. Irvine, Feb 29 2020