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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)
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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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(n)= 2*a(n-2) - a(n-4). - Paul Curtz, Mar 09 2011
a(n) = n + a(n-1) for odd n; a(n) = n - A064455(n-1) for even n.
a(n) = Sum_{k=1..n} (1 + (k-1)*(-1)^(k-1)). - Bruno Berselli, Jul 16 2013
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EXAMPLE
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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
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MAPLE
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MATHEMATICA
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CoefficientList[Series[(1 +x +2*x^2)/((1-x)^2*(1+x)^2), {x, 0, 70}], x] (* Wesley Ivan Hurt, Jul 26 2014 *)
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PROG
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(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]
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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