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%I #46 Mar 01 2020 02:09:43
%S 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,
%T 14,28,15,30,16,32,17,34,18,36,19,38,20,40,21,42,22,44,23,46,24,48,25,
%U 50,26,52,27,54,28,56,29,58,30,60,31,62,32,64,33,66,34,68,35,70,36,72
%N The natural numbers interleaved with the even numbers.
%C a(n) = number of ordered, length two, compositions of n with at least one odd summand - _Len Smiley_, Nov 25 2001
%C Also number of 0's in n-th row of triangle in A071037. - _Hans Havermann_, May 26 2002
%C a(n) = (n - n mod 2)/(2 - n mod 2). - _Reinhard Zumkeller_, Jul 30 2002
%C For n > 2: a(n) = number of odd terms in row n-2 of triangle A265705. - _Reinhard Zumkeller_, Dec 15 2015
%H Reinhard Zumkeller, <a href="/A029578/b029578.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F a(n) = (3*n/2-1+(1-n/2)*(-1)^n)/2. a(n+4)=2*a(n+2)-a(n).
%F G.f.: x^2*(2x+1)/(1-x^2)^2; a(n)=floor((n+1)/2)+(n is odd)*floor((n+1)/2)
%F a(n) = floor(n/2)*binomial(2, mod(n, 2)) - _Paul Barry_, May 25 2003
%F a(2*n) = n, a(2*n-1) = 2*n-2. a(-n)=-A065423(n+2).
%F 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
%F a(n) = Sum_{k=0..n-2} gcd(n-k-1, k+1). - _Paul Barry_, May 03 2005
%F For n>6: a(n) = floor(a(n-1)*a(n-2)/a(n-3)). [_Reinhard Zumkeller_, Mar 06 2011]
%t 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 *)
%o (PARI) a(n)=if(n%2,n-1,n/2)
%o (Haskell)
%o import Data.List (transpose)
%o a029578 n = (n - n `mod` 2) `div` (2 - n `mod` 2)
%o a029578_list = concat $ transpose [a001477_list, a005843_list]
%o -- _Reinhard Zumkeller_, Nov 27 2012
%Y Cf. A065423 (at least one even summand).
%Y Cf. A009531.
%Y Cf. A001477, A005843, A211538 (partial sums).
%Y Cf. A265705.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_.
%E Explicated definition by _Reinhard Zumkeller_, Nov 27 2012
%E Title simplified by _Sean A. Irvine_, Feb 29 2020
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