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A029578 The natural numbers interleaved with the even numbers.

%I A029578 #49 Jan 23 2025 00:15:45
%S A029578 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 A029578 14,28,15,30,16,32,17,34,18,36,19,38,20,40,21,42,22,44,23,46,24,48,25,
%U A029578 50,26,52,27,54,28,56,29,58,30,60,31,62,32,64,33,66,34,68,35,70,36,72
%N A029578 The natural numbers interleaved with the even numbers.
%C A029578 a(n) = number of ordered, length two, compositions of n with at least one odd summand - _Len Smiley_, Nov 25 2001
%C A029578 Also number of 0's in n-th row of triangle in A071037. - _Hans Havermann_, May 26 2002
%C A029578 For n > 2: a(n) = number of odd terms in row n-2 of triangle A265705. - _Reinhard Zumkeller_, Dec 15 2015
%H A029578 Reinhard Zumkeller, <a href="/A029578/b029578.txt">Table of n, a(n) for n = 0..10000</a>
%H A029578 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A029578 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A029578 a(n) = (3*n - 2 - (-1)^n*(n - 2))/4.
%F A029578 a(n+4) = 2*a(n+2) - a(n).
%F A029578 G.f.: x^2*(1 + 2*x)/(1-x^2)^2.
%F A029578 a(n) = floor((n+1)/2) + (n is odd)*floor((n+1)/2).
%F A029578 a(n) = (n - n mod 2)/(2 - n mod 2). - _Reinhard Zumkeller_, Jul 30 2002
%F A029578 a(n) = floor(n/2)*binomial(2, mod(n, 2)) - _Paul Barry_, May 25 2003
%F A029578 a(2*n) = n.
%F A029578 a(2*n-1) = 2*n-2.
%F A029578 a(-n) = -A065423(n+2).
%F A029578 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 A029578 a(n) = Sum_{k=0..n-2} gcd(n-k-1, k+1). - _Paul Barry_, May 03 2005
%F A029578 For n>6: a(n) = floor(a(n-1)*a(n-2)/a(n-3)). - _Reinhard Zumkeller_, Mar 06 2011
%F A029578 E.g.f.: (1/4)*((x+2)*exp(-x) + (3*x-2)*exp(x)). - _G. C. Greubel_, Jan 22 2025
%t A029578 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 A029578 (PARI) a(n)=if(n%2,n-1,n/2)
%o A029578 (Haskell)
%o A029578 import Data.List (transpose)
%o A029578 a029578 n =  (n - n `mod` 2) `div` (2 - n `mod` 2)
%o A029578 a029578_list = concat $ transpose [a001477_list, a005843_list]
%o A029578 -- _Reinhard Zumkeller_, Nov 27 2012
%o A029578 (Magma)
%o A029578 A029578:= func< n | (n + (n-2)*(n mod 2))/2 >;
%o A029578 [A029578(n): n in [0..80]]; // _G. C. Greubel_, Jan 22 2025
%o A029578 (Python)
%o A029578 def A029578(n): return (n + (n-2)*(n%2))//2
%o A029578 print([A029578(n) for n in range(81)]) # _G. C. Greubel_, Jan 22 2025
%Y A029578 Cf. A065423 (at least one even summand).
%Y A029578 Cf. A001477, A005843, A009531, A071037, A211538 (partial sums), A265705.
%K A029578 nonn,easy
%O A029578 0,4
%A A029578 _N. J. A. Sloane_
%E A029578 Explicated definition by _Reinhard Zumkeller_, Nov 27 2012
%E A029578 Title simplified by _Sean A. Irvine_, Feb 29 2020