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%I #30 Jun 12 2022 08:19:18
%S 1,1,2,3,4,6,8,11,14,18,22,28,34,42,50,61,72,86,100,118,136,158,180,
%T 208,236,270,304,346,388,438,488,549,610,682,754,840,926,1026,1126,
%U 1244,1362,1498,1634,1792,1950,2130,2310,2518,2726,2962,3198,3468,3738,4042
%N a(1)=1, a(n) = Sum_{k=1..n-1} min(a(k), a(n-k)).
%C Sequence gives 1/2 of the number of unique path partitions of the integer 2n; see the function w(n) as defined in the paper by Bessenrodt, Olsson, and Sellers.
%H T. D. Noe, <a href="/A075535/b075535.txt">Table of n, a(n) for n = 1..10000</a>
%H Christine Bessenrodt, Jørn B. Olsson, and James A. Sellers, <a href="http://arxiv.org/abs/1107.1156">Unique path partitions: Characterization and Congruences</a>, arXiv:1107.1156 [math.CO], 2011-2012.
%F a(1)=a(2)=1; a(2n) = a(2n-1) + a(n); a(2n+1) = a(2n) + a(n); for n >= 3, a(n) = a(n-1) + a(floor(n/2)).
%F Let T(x) be the g.f. 1 + x + 2*x^2 + 3*x^3 + ... (i.e., with offset 0), then T(x) = 1 + x * (1+x)/(1-x) * T(x^2). - _Joerg Arndt_, May 11 2010
%t Fold[Append[#1, Total[Take[Flatten[Transpose[{#1, #1}]], #2]]] &, {1}, Range[53]] (* _Birkas Gyorgy_, Apr 18 2011 *)
%t a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 1] + a[Floor[n/2]]; Array[a, 100] (* _T. D. Noe_, Apr 18 2011 *)
%o (PARI) a(n)=if(n<3,1,a(n-1)+a(floor(n/2)))
%Y Cf. A033485.
%K nonn
%O 1,3
%A _Benoit Cloitre_, Jan 11 2003
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