A029578 - OEIS

%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