

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 >= 2.


1



1, 1, 3, 4, 7, 11, 19, 29, 47, 76, 125, 200, 322, 519, 845, 1366, 2211, 3573, 5778, 9342, 15122, 24481, 39639, 64094, 103684, 167734, 271397, 439178, 710698, 1149964, 1860751, 3010500, 4870792, 7880666, 12751729, 20632965, 33385273, 54019297, 87406719
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OFFSET

2,3


REFERENCES

The question of enumerating these partitions appears as Problem 11005, American Mathematical Monthly, 110, April 2003, page 340.
Problem 11005, American Math. Monthly, Vol. 112, 2005, pp. 8990. (The published solution is incomplete; the solver's expression q_2(n,d) must be summed over all d = 1,2,...,floor{n/2}.)


LINKS

T. D. Noe, Table of n, a(n) for n=2..500
Marty Getz and Dixon Jones, Problem 11005, American Mathematical Monthly, 110, April 2003, page 340.
Marty Getz, Dixon Jones and Ken Dutch, Partitioning by Arithmetic Progressions: 11005, American Math. Monthly, Vol. 112, 2005, pp. 8990.


FORMULA

a(n) = sum_{d=1}^{floor{n/2}} {{F_k}^r}*{F_{k1}}^{dr}, where d is the common difference of the arithmetic progressions, k = Floor{n/d}, r = n mod d and F_k is the kth Fibonacci number (A000045).  Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu), May 21 2005


EXAMPLE

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}.


PROG

(PARI) a(n) = sum(d = 1, n\2, fibonacci(n\d)^(n % d) * fibonacci(n\d 1)^(d  n%d)); \\ Michel Marcus, Oct 13 2013


CROSSREFS

A053732 relates to partitions of {1, 2, ..., n} into arithmetic progressions without restrictions on the common difference of the progressions.
Sequence in context: A041739 A042593 A041018 * A049863 A025068 A049928
Adjacent sequences: A072252 A072253 A072254 * A072256 A072257 A072258


KEYWORD

easy,nice,nonn


AUTHOR

Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu), Jul 08 2002


EXTENSIONS

More terms from Michel Marcus, Oct 13 2013


STATUS

approved



