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A123684
Alternate A016777(n) with A000027(n).
13
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
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
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
KEYWORD
nonn,easy,changed
AUTHOR
Alford Arnold, Oct 11 2006
EXTENSIONS
More terms from Klaus Brockhaus, May 12 2007
STATUS
approved