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A342194 Number of strict compositions of n with equal differences, or strict arithmetic progressions summing to n.

%I A342194 #7 Apr 02 2021 08:35:38
%S A342194 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 A342194 29,45,33,29,55,31,35,55,39,43,65,37,43,65,51,41,77,43,51,85,53,47,85,
%U A342194 53,65,87,61,53,99,67,67,97,67,59,119,61,71,113,75,79,123,67,79,117
%N A342194 Number of strict compositions of n with equal differences, or strict arithmetic progressions summing to n.
%H A342194 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>.
%H A342194 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%F A342194 a(n > 0) = A175342(n) - A000005(n) + 1.
%F A342194 a(n > 0) = 2*A049988(n) - 2*A000005(n) + 1 = 2*A049982(n) + 1.
%e A342194 The a(1) = 1 through a(9) = 13 compositions:
%e A342194   (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
%e A342194             (1,2)  (1,3)  (1,4)  (1,5)    (1,6)  (1,7)  (1,8)
%e A342194             (2,1)  (3,1)  (2,3)  (2,4)    (2,5)  (2,6)  (2,7)
%e A342194                           (3,2)  (4,2)    (3,4)  (3,5)  (3,6)
%e A342194                           (4,1)  (5,1)    (4,3)  (5,3)  (4,5)
%e A342194                                  (1,2,3)  (5,2)  (6,2)  (5,4)
%e A342194                                  (3,2,1)  (6,1)  (7,1)  (6,3)
%e A342194                                                         (7,2)
%e A342194                                                         (8,1)
%e A342194                                                         (1,3,5)
%e A342194                                                         (2,3,4)
%e A342194                                                         (4,3,2)
%e A342194                                                         (5,3,1)
%t A342194 Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],SameQ@@Differences[#]&]],{n,0,30}]
%Y A342194 Strict compositions in general are counted by A032020.
%Y A342194 The unordered version is A049980.
%Y A342194 The non-strict version is A175342.
%Y A342194 A000203 adds up divisors.
%Y A342194 A000726 counts partitions with alternating parts unequal.
%Y A342194 A003242 counts anti-run compositions.
%Y A342194 A224958 counts compositions with alternating parts unequal.
%Y A342194 A342343 counts compositions with alternating parts strictly decreasing.
%Y A342194 A342495 counts compositions with constant quotients.
%Y A342194 A342527 counts compositions with alternating parts equal.
%Y A342194 Cf. A000009, A001522, A002843, A049988, A062968, A070211, A114921, A325545, A325557, A342496, A342515, A342522.
%K A342194 nonn
%O A342194 0,4
%A A342194 _Gus Wiseman_, Apr 02 2021