cp's OEIS Frontend

A349801 Number of integer partitions of n into three or more parts or into two equal parts.

%I A349801 #6 Dec 25 2021 02:44:58
%S A349801 0,0,1,1,3,4,8,11,18,25,37,50,71,94,128,168,223,288,376,480,617,781,
%T A349801 991,1243,1563,1945,2423,2996,3704,4550,5589,6826,8333,10126,12293,
%U A349801 14865,17959,21618,25996,31165,37318,44562,53153,63239,75153,89111,105535,124730
%N A349801 Number of integer partitions of n into three or more parts or into two equal parts.
%C A349801 This sequence arose as the following degenerate case. If we define a sequence to be alternating if it is alternately strictly increasing and strictly decreasing, starting with either, then a(n) is the number of non-alternating integer partitions of n. Under this interpretation:
%C A349801 - The non-strict case is A047967, weak A349796, weak complement A349795.
%C A349801 - The complement is counted by A065033(n) = ceiling(n/2) for n > 0.
%C A349801 - These partitions are ranked by A289553 \ {1}, complement A167171 \/ {1}.
%C A349801 - The version for compositions is A345192, ranked by A345168.
%C A349801 - The weak version for compositions is A349053, ranked by A349057.
%C A349801 - The weak version is A349061, complement A349060, ranked by A349794.
%F A349801 a(1) = 0; a(n > 0) = A000041(n) - ceiling(n/2).
%e A349801 The a(2) = 1 through a(7) = 11 partitions:
%e A349801   (11)  (111)  (22)    (221)    (33)      (322)
%e A349801                (211)   (311)    (222)     (331)
%e A349801                (1111)  (2111)   (321)     (421)
%e A349801                        (11111)  (411)     (511)
%e A349801                                 (2211)    (2221)
%e A349801                                 (3111)    (3211)
%e A349801                                 (21111)   (4111)
%e A349801                                 (111111)  (22111)
%e A349801                                           (31111)
%e A349801                                           (211111)
%e A349801                                           (1111111)
%t A349801 Table[Length[Select[IntegerPartitions[n],MatchQ[#,{x_,x_}|{_,_,__}]&]],{n,0,10}]
%Y A349801 A000041 counts partitions, ordered A011782.
%Y A349801 A001250 counts alternating permutations, complement A348615.
%Y A349801 A004250 counts partitions into three or more parts, strict A347548.
%Y A349801 A025047/A025048/A025049 count alternating compositions, ranked by A345167.
%Y A349801 A096441 counts weakly alternating 0-appended partitions.
%Y A349801 A345165 counts partitions w/ no alternating permutation, complement A345170.
%Y A349801 Cf. A000070, A001700, A002865, A117298, A117989, A102726, A128761, A345162, A345163, A345166, A349798.
%K A349801 nonn
%O A349801 0,5
%A A349801 _Gus Wiseman_, Dec 23 2021