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); 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
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
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Jan 11 2003
STATUS
approved