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A075535
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a(1)=1, a(n) = Sum_{k=1..n-1} min(a(k), a(n-k)).
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3
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1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 22, 28, 34, 42, 50, 61, 72, 86, 100, 118, 136, 158, 180, 208, 236, 270, 304, 346, 388, 438, 488, 549, 610, 682, 754, 840, 926, 1026, 1126, 1244, 1362, 1498, 1634, 1792, 1950, 2130, 2310, 2518, 2726, 2962, 3198, 3468, 3738, 4042
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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)).
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
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MATHEMATICA
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Fold[Append[#1, Total[Take[Flatten[Transpose[{#1, #1}]], #2]]] &, {1}, Range[53]] (* Birkas Gyorgy, Apr 18 2011 *)
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 *)
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PROG
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(PARI) a(n)=if(n<3, 1, a(n-1)+a(floor(n/2)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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