Number of distinct sets of nonnegative integers with perimeter n, as defined in the comments.

Terms

a(0) =2a(1) =2a(2) =3a(3) =4a(4) =6a(5) =8a(6) =11a(7) =14a(8) =19a(9) =24a(10) =31a(11) =39a(12) =50a(13) =61a(14) =77a(15) =94a(16) =117a(17) =141a(18) =173a(19) =208a(20) =253a(21) =302a(22) =363a(23) =431a(24) =516a(25) =609a(26) =723a(27) =850a(28) =1003a(29) =1174