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%I #40 Dec 17 2022 08:22:28
%S 0,1,2,2,4,3,4,4,8,5,6,6,8,7,8,8,16,9,10,10,12,11,12,12,16,13,14,14,
%T 16,15,16,16,32,17,18,18,20,19,20,20,24,21,22,22,24,23,24,24,32,25,26,
%U 26,28,27,28,28,32,29,30,30,32,31,32,32,64,33,34,34,36,35,36,36,40,37,38
%N a(n) = a(n - a(n-1)) + a(floor(n/2)).
%C From _M. F. Hasler_, Oct 20 2019: (Start)
%C The sequence A285326/2 is characterized by a(2n) = 2*a(n) (n >= 0) and a(2n-1) = n (n >= 1). This implies the property defining this sequence: If n = 2k, then n - a(n-1) = 2k - a(2k-1) = 2k - k = k, so a(n - a(n-1)) + a(floor(n/2)) = a(k) + a(k) = 2*a(k) = a(2k) = a(n). If n = 2k-1, then n - a(n-1) = 2k-1 - a(2k-2) = 2k-1 - 2*a(k-1), whence a(n - a(n-1)) + a(floor(n/2)) = a(2(k - a(k-1)) - 1) + a(k-1) = k - a(k-1) + a(k-1) = k = a(2k-1) = a(n). Thus, A285326/2 satisfies the definition of this sequence.
%C The sequence is equal to itself multiplied by 2 and interleaved with the positive integers. (This is equivalent to the above characterization.)
%C The sequence repeats the pattern [A, B, C, C] where in the n-th occurrence C = 2n, B = C - 1, A = C if n is even, A = C + 2 if n == 3 (mod 4), and A = 16*a((n-1)/4) otherwise. This yields a simpler formula for all terms except for indices which are multiples of 16. (End)
%H Reinhard Zumkeller, <a href="/A140472/b140472.txt">Table of n, a(n) for n = 0..10000</a>
%F a(0) = 0; a(1) = a(2) = 1; a(n) = a(n - a(n-1)) + a(floor(n/2)).
%F a(n) = (n+A006519(n))/2 for n > 0 (conjectured). - _Jon Maiga_, Aug 16 2019
%F a(n) = A285326(n)/2, equivalent to the above: see comments for the proof. - _M. F. Hasler_, Oct 19 2019
%t a[0] = 0; a[1] = 1;
%t a[n_] := a[n] = a[n - a[n - 1]] + a[Floor[n/2]];
%t Table[a[n], {n, 0, 200}]
%o (Haskell)
%o a140472 n = a140472_list !! n
%o a140472_list = 0 : 1 : h 2 1 where
%o h x y = z : h (x + 1) z where z = a140472 (x - y) + a140472 (x `div` 2)
%o -- _Reinhard Zumkeller_, Jul 20 2012
%o (Magma) I:=[1,2]; [0] cat [n le 2 select I[n] else Self(n-Self(n-1))+Self(Floor((n) div 2)):n in [1..75]]; // _Marius A. Burtea_, Aug 16 2019
%o (PARI) a(n)=(n+bitand(n,-n))\2 \\ _M. F. Hasler_, Oct 19 2019
%Y Cf. A004001, A006519.
%Y Cf. A214546 (first differences).
%Y Same as A109168, if a(0) = 0 is omitted. - _M. F. Hasler_, Oct 19 2019
%K nonn
%O 0,3
%A _Roger L. Bagula_ and _Gary W. Adamson_, Jun 28 2008
%E Offset corrected by _Reinhard Zumkeller_, Jul 20 2012
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