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A029578
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The natural numbers interleaved with the even numbers.
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21
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0, 0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, 6, 12, 7, 14, 8, 16, 9, 18, 10, 20, 11, 22, 12, 24, 13, 26, 14, 28, 15, 30, 16, 32, 17, 34, 18, 36, 19, 38, 20, 40, 21, 42, 22, 44, 23, 46, 24, 48, 25, 50, 26, 52, 27, 54, 28, 56, 29, 58, 30, 60, 31, 62, 32, 64, 33, 66, 34, 68, 35, 70, 36, 72
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OFFSET
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0,4
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COMMENTS
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a(n) = number of ordered, length two, compositions of n with at least one odd summand - Len Smiley, Nov 25 2001
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LINKS
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FORMULA
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a(n) = (3*n/2-1+(1-n/2)*(-1)^n)/2. a(n+4)=2*a(n+2)-a(n).
G.f.: x^2*(2x+1)/(1-x^2)^2; a(n)=floor((n+1)/2)+(n is odd)*floor((n+1)/2)
a(n) = floor(n/2)*binomial(2, mod(n, 2)) - Paul Barry, May 25 2003
a(2*n) = n, a(2*n-1) = 2*n-2. a(-n)=-A065423(n+2).
a(n) = Sum_{k=0..floor((n-2)/2)} (C(n-k-2, k) mod 2)((1+(-1)^k)/2)*2^A000120(n-2k-2). - Paul Barry, Jan 06 2005
a(n) = Sum_{k=0..n-2} gcd(n-k-1, k+1). - Paul Barry, May 03 2005
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MATHEMATICA
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With[{nn=40}, Riffle[Range[0, nn], Range[0, 2nn, 2]]] (* or *) LinearRecurrence[ {0, 2, 0, -1}, {0, 0, 1, 2}, 80] (* Harvey P. Dale, Aug 23 2015 *)
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PROG
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(PARI) a(n)=if(n%2, n-1, n/2)
(Haskell)
import Data.List (transpose)
a029578 n = (n - n `mod` 2) `div` (2 - n `mod` 2)
a029578_list = concat $ transpose [a001477_list, a005843_list]
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CROSSREFS
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Cf. A065423 (at least one even summand).
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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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