This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A070211 #25 Dec 07 2019 12:18:23
%S A070211 1,1,2,4,6,9,14,18,24,34,42,52,68,82,101,126,147,175,213,246,289,344,
%T A070211 392,453,530,598,687,791,885,1007,1151,1276,1438,1629,1806,2018,2262,
%U A070211 2490,2775,3091,3387,3754,4165,4542,5011,5527,6012,6600,7245,7864,8614
%N A070211 Number of compositions (ordered partitions) of n that are concave-down sequences.
%C A070211 Here, a finite sequence is concave if each term (other than the first or last) is at least the average of the two adjacent terms. - _Eric M. Schmidt_, Sep 29 2013
%C A070211 The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1). Then a(n) is the number of compositions of n with weakly decreasing differences. - _Gus Wiseman_, May 15 2019
%H A070211 Alois P. Heinz, <a href="/A070211/b070211.txt">Table of n, a(n) for n = 0..500</a>
%H A070211 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A070211 Out of the 8 ordered partitions of 4, only 2+1+1 and 1+1+2 are not concave, so a(4)=6.
%e A070211 From _Gus Wiseman_, May 15 2019: (Start)
%e A070211 The a(1) = 1 through a(6) = 14 compositions:
%e A070211 (1) (2) (3) (4) (5) (6)
%e A070211 (11) (12) (13) (14) (15)
%e A070211 (21) (22) (23) (24)
%e A070211 (111) (31) (32) (33)
%e A070211 (121) (41) (42)
%e A070211 (1111) (122) (51)
%e A070211 (131) (123)
%e A070211 (221) (132)
%e A070211 (11111) (141)
%e A070211 (222)
%e A070211 (231)
%e A070211 (321)
%e A070211 (1221)
%e A070211 (111111)
%e A070211 (End)
%t A070211 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@Differences[#]&]],{n,0,15}] (* _Gus Wiseman_, May 15 2019 *)
%o A070211 (Sage) def A070211(n) : return sum(all(2*p[i] >= p[i-1] + p[i+1] for i in range(1, len(p)-1)) for p in Compositions(n)) # _Eric M. Schmidt_, Sep 29 2013
%Y A070211 Cf. A000079, A001523 (weakly unimodal compositions), A069916, A175342, A320466, A325361 (concave-down partitions), A325545, A325546 (concave-up compositions), A325547, A325548, A325557.
%K A070211 nice,nonn
%O A070211 0,3
%A A070211 _Pontus von Brömssen_, May 07 2002
%E A070211 Name edited by _Gus Wiseman_, May 15 2019