%I #10 Mar 24 2021 22:19:32

%S 1,1,2,4,6,8,11,12,16,17,21,20,29,24,31,32,38,32,46,36,51,46,51,44,69,

%T 51,61,60,73,56,87,60,84,74,81,76,110,72,91,88,115,80,123,84,117,112,

%U 111,92,153,101,132,116,139,104,159,120,161,130,141,116,205,120,151,156,178,142,195,132,183,158

%N Number of compositions of n with alternating parts equal.

%C These are finite sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i.

%H Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.

%F a(n) = 1 + n + A000203(n) - 2*A000005(n).

%F a(n) = A065608(n) + A062968(n).

%e The a(1) = 1 through a(8) = 16 compositions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (12) (13) (14) (15) (16) (17)

%e (21) (22) (23) (24) (25) (26)

%e (111) (31) (32) (33) (34) (35)

%e (121) (41) (42) (43) (44)

%e (1111) (131) (51) (52) (53)

%e (212) (141) (61) (62)

%e (11111) (222) (151) (71)

%e (1212) (232) (161)

%e (2121) (313) (242)

%e (111111) (12121) (323)

%e (1111111) (1313)

%e (2222)

%e (3131)

%e (21212)

%e (11111111)

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Plus@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]

%Y The odd-length case is A062968.

%Y The even-length case is A065608.

%Y The version with alternating parts unequal is A224958 (unordered: A000726).

%Y The version with alternating parts weakly decreasing is A342528.

%Y A000005 counts constant compositions.

%Y A000041 counts weakly increasing (or weakly decreasing) compositions.

%Y A000203 adds up divisors.

%Y A002843 counts compositions with all adjacent parts x <= 2y.

%Y A003242 counts anti-run compositions.

%Y A175342 counts compositions with constant differences.

%Y A342495 counts compositions with constant first quotients.

%Y A342496 counts partitions with constant first quotients (strict: A342515, ranking: A342522).

%Y Cf. A001522, A008965, A048004, A059966, A064410, A064428, A069916, A114921, A167606, A325545, A325557.

%K nonn

%O 0,3

%A _Gus Wiseman_, Mar 24 2021