%I A072255
%S A072255 1,1,3,4,7,11,19,29,47,76,125,200,322,519,845,1366,2211,3573,5778,9342,
%T A072255 15122,24481,39639,64094,103684,167734,271397,439178,710698
%N A072255 Number of ways to partition {1,2,...,n} into arithmetic progressions, 
               where in each partition all the progressions have the same common 
               difference and have lengths greater than or equal to 2.
%D A072255 The question of enumerating these partitions appears as Problem 11005, 
               American Mathematical Monthly, 110, April 2003, page 340.
%D A072255 Problem 11005, American Math. Monthly, Vol. 112, 2005, pp. 89-90. (The 
               published solution is incomplete; the solver's expression q_2(n,d) 
               must be summed over all d = 1,2,...,floor{n/2}.)
%H A072255 T. D. Noe, <a href="b072255.txt">Table of n, a(n) for n=2..500</a>
%F A072255 a(n) = sum_{d=1}^{floor{n/2}} {{F_k}^r}*{F_{k-1}}^{d-r}, where d is the 
               common difference of the arithmetic progressions, k = Floor{n/d}, 
               r = n mod d and F_k is the k-th Fibonacci number (A000045). - Marty 
               Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu), May 
               21 2005
%e A072255 a(5)=4: the four ways to partition {1,2,3,4,5} as described above are: 
               {1,2}{3,4,5}; {1,2,3}{4,5}; {1,2,3,4,5}; {1,3,5}{2,4}.
%Y A072255 A053732 relates to partitions of {1, 2, ..., n} into arithmetic progressions 
               without restrictions on the common difference of the progressions.
%Y A072255 Sequence in context: A041739 A042593 A041018 this_sequence A049863 A025068 
               A049928
%Y A072255 Adjacent sequences: A072252 A072253 A072254 this_sequence A072256 A072257 
               A072258
%K A072255 easy,nice,nonn
%O A072255 2,3
%A A072255 Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu) (fndjj(AT)uaf.edu), 
               Jul 08 2002