|
%I #7 Apr 02 2021 08:35:38
%S 1,1,1,3,3,5,7,7,7,13,11,11,17,13,15,25,17,17,29,19,23,35,25,23,39,29,
%T 29,45,33,29,55,31,35,55,39,43,65,37,43,65,51,41,77,43,51,85,53,47,85,
%U 53,65,87,61,53,99,67,67,97,67,59,119,61,71,113,75,79,123,67,79,117
%N Number of strict compositions of n with equal differences, or strict arithmetic progressions summing to n.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>.
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%F a(n > 0) = A175342(n) - A000005(n) + 1.
%F a(n > 0) = 2*A049988(n) - 2*A000005(n) + 1 = 2*A049982(n) + 1.
%e The a(1) = 1 through a(9) = 13 compositions:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8)
%e (2,1) (3,1) (2,3) (2,4) (2,5) (2,6) (2,7)
%e (3,2) (4,2) (3,4) (3,5) (3,6)
%e (4,1) (5,1) (4,3) (5,3) (4,5)
%e (1,2,3) (5,2) (6,2) (5,4)
%e (3,2,1) (6,1) (7,1) (6,3)
%e (7,2)
%e (8,1)
%e (1,3,5)
%e (2,3,4)
%e (4,3,2)
%e (5,3,1)
%t Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],SameQ@@Differences[#]&]],{n,0,30}]
%Y Strict compositions in general are counted by A032020.
%Y The unordered version is A049980.
%Y The non-strict version is A175342.
%Y A000203 adds up divisors.
%Y A000726 counts partitions with alternating parts unequal.
%Y A003242 counts anti-run compositions.
%Y A224958 counts compositions with alternating parts unequal.
%Y A342343 counts compositions with alternating parts strictly decreasing.
%Y A342495 counts compositions with constant quotients.
%Y A342527 counts compositions with alternating parts equal.
%Y Cf. A000009, A001522, A002843, A049988, A062968, A070211, A114921, A325545, A325557, A342496, A342515, A342522.
%K nonn
%O 0,4
%A _Gus Wiseman_, Apr 02 2021
|
