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A075535 a(1)=1, a(n) = sum_{k=1..n-1} min(a(k), a(n-k)). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Christine Bessenrodt, Jørn B. Olsson, and James A. Sellers, Unique path partitions:  Characterization and Congruences, arXiv:1107.1156 [math.CO], 2011-2012.

FORMULA

a(1)=a(2)=1; a(2n)=a(2n-1)+a(n); a(2n+1)=a(2n)+a(n); 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

MATHEMATICA

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 *)

PROG

(PARI) a(n)=if(n<3, 1, a(n-1)+a(floor(n/2)))

CROSSREFS

Cf. A033485.

Sequence in context: A290743 A059291 A177339 * A238383 A134953 A175870

Adjacent sequences:  A075532 A075533 A075534 * A075536 A075537 A075538

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Jan 11 2003

STATUS

approved

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Last modified May 31 09:30 EDT 2020. Contains 334748 sequences. (Running on oeis4.)