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

Author

Alford Arnold, Oct 11 2006

Keywords

• nonn
• easy

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

Images under the modified '3x-1' map: a(n) = n/2 if n is even, (3n-1)/2 if n is odd. (In this sequence, the numbers at even indices n are n/2 [A000027], and the numbers at odd indices n are 3((n-1)/2) + 1 [A016777] = (3n-1)/2.) The latter correspondence interestingly mirrors an insight in David Bařina's 2020 paper (see below), namely that 3(n+1)/2 - 1 = (3n+1)/2. - Kevin Ge, Oct 30 2024

Examples

			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
		

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).
• David Bařina, Convergence verification of the Collatz problem

Crossrefs

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

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

Cf. A014682 (modified Collatz map)

Programs

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

&cat[ [ 3*n-2, n ]: n in [1..36] ]; // Klaus Brockhaus, May 12 2007

Magma

/* From the fourteenth formula: */ [&+[1+k*(-1)^k: k in [0..n]]: n in [0..80]]; // Bruno Berselli, Jul 16 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 *)
LinearRecurrence[{0,2,0,-1},{1,1,4,2},80] (* Harvey P. Dale, Apr 14 2025 *)

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

SageMath

[(n + (2*n-1)*(n%2))//2 for n in range(1,71)] # G. C. Greubel, Mar 15 2024

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

Extensions

More terms from Klaus Brockhaus, May 12 2007