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A123684 Alternate A016777(n) with A000027(n). 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n) is a diagonal of Table A123685.

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

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) = n - 1/4 - (1/2*n - 1/4)*(-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(1+2*n)+a(2+2*n) = 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) = sum(1+(k-1)*(-1)^(k-1), k=1..n). - 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) 1. print(vector(72, n, if(n%2==0, n/2, (3*n-1)/2))); 2. 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

CROSSREFS

Cf. A000027, A016777, A123685, A071045, A093005, A105638, A092530, A031878.

Sequence in context: A026189 A026213 A225126 * A180076 A169756 A002949

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

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Last modified June 15 17:43 EDT 2019. Contains 324142 sequences. (Running on oeis4.)