A123684 Alternate A016777(n) with A000027(n).

1, 1, 4, 2, 7, 3, 10, 4, 13, 5, 16, 6, 19, 7, 22, 8, 25, 9, 28, 10, 31, 11, 34, 12, 37, 13, 40, 14, 43, 15, 46, 16, 49, 17, 52, 18, 55, 19, 58, 20, 61, 21, 64, 22, 67, 23, 70, 24, 73, 25, 76, 26, 79, 27, 82, 28, 85, 29, 88, 30, 91, 31, 94, 32, 97, 33, 100, 34, 103, 35, 106, 36

Offset

1,3

Comments

a(n) is a diagonal of Table A123685.

The arithmetic average of the first n terms gives the positive integers repeated (A008619). - Philippe Deléham, Nov 20 2013

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Formula

From Klaus Brockhaus, May 12 2007: (Start)

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

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

a(2n-1) = A016777(n-1) = 3(n-1) + 1.

a(2n) = A000027(n) = n.

a(n) = A071045(n-1) + 1.

a(n) = A093005(n) - A093005(n-1) for n > 1.

a(n) = A105638(n+2) - A105638(n+1) for n > 1.

a(n) = A092530(n) - A092530(n-1) - 1.

a(n) = A031878(n+1) - A031878(n) - 1. (End)

a(2*n+1) + a(2*n+2) = A016825(n). - Paul Curtz, Mar 09 2011

a(n)= 2*a(n-2) - a(n-4). - Paul Curtz, Mar 09 2011

From Jaroslav Krizek, Mar 22 2011 (Start):

a(n) = n + a(n-1) for odd n; a(n) = n - A064455(n-1) for even n.

a(n) = A064455(n) - A137501(n).

Abs(a(n) - A064455(n)) = A052928(n). (End)

a(n) = A225126(n) for n > 1. - Reinhard Zumkeller, Apr 29 2013

a(n) = Sum_{k=1..n} (1 + (k-1)*(-1)^(k-1)). - Bruno Berselli, Jul 16 2013

a(n) = n + floor(n/2) for odd n; a(n) = n/2 for even n. - Reinhard Muehlfeld, Jul 25 2014

Example

The natural numbers begin 1, 2, 3, ... (A000027), the sequence 3*n + 1 begins 1, 4, 7, 10, ... (A016777), therefore A123684 begins 1, 1, 4, 2, 7, 3, 10, ...
1/1 = 1, (1+1)/2 = 1, (1+1+4)/3 = 2, (1+1+4+2)/4 = 2, ... - Philippe Deléham, Nov 20 2013

Maple

A123684:=n->n-1/4-(1/2*n-1/4)*(-1)^n: seq(A123684(n), n=1..70); # Wesley Ivan Hurt, Jul 26 2014

Mathematica

CoefficientList[Series[(1 +x +2*x^2)/((1-x)^2*(1+x)^2), {x, 0, 70}], x] (* Wesley Ivan Hurt, Jul 26 2014 *)

Prog

(Magma) &cat[ [ 3*n-2, n ]: n in [1..36] ]; // Klaus Brockhaus, May 12 2007
(PARI) print(vector(72, n, if(n%2==0, n/2, (3*n-1)/2))) \\ Klaus Brockhaus, May 12 2007
(PARI) print(vector(72, n, n-1/4-(1/2*n-1/4)*(-1)^n)); \\ Klaus Brockhaus, May 12 2007
(Haskell)
import Data.List (transpose)
a123684 n = a123684_list !! (n-1)
a123684_list = concat $ transpose [a016777_list, a000027_list]
-- Reinhard Zumkeller, Apr 29 2013
(Magma) /* From the fourteenth formula: */ [&+[1+k*(-1)^k: k in [0..n]]: n in [0..80]]; // Bruno Berselli, Jul 16 2013
(SageMath) [(n + (2*n-1)*(n%2))//2 for n in range(1, 71)] # G. C. Greubel, Mar 15 2024

Crossrefs

Cf. A000027, A008619, A016777, A016825, A031878, A052928, A064455.

Cf. A071045, A092530, A093005, A105638, A123685, A137501, A225126.

Sequence in context: A026213 A319556 A225126 * A180076 A169756 A329796

Adjacent sequences: A123681 A123682 A123683 * A123685 A123686 A123687

Keyword

nonn,easy

Author

Alford Arnold, Oct 11 2006

Extensions

More terms from Klaus Brockhaus, May 12 2007

Status

approved