|
%I #15 Feb 21 2023 13:23:31
%S 1,1,2,4,5,6,8,10,10,11,12,12,16,16,18,20,19,18,22,22,24,28,24,24,30,
%T 27,30,30,34,30,38,36,36,36,36,40,43,40,42,46,48,42,52,46,48,52,48,48,
%U 56,55,54,54,58,54,60,58,64,64,60,60,72,64,68,74,69,72,72
%N Number of compositions of n with constant (equal) first quotients.
%C The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%F a(n > 0) = 2*A342496(n) - A000005(n).
%e The composition (1,2,4,8) has first quotients (2,2,2) so is counted under a(15).
%e The composition (4,5,6) has first quotients (5/4,6/5) so is not counted under a(15).
%e The a(1) = 1 through a(7) = 10 compositions:
%e (1) (2) (3) (4) (5) (6) (7)
%e (11) (12) (13) (14) (15) (16)
%e (21) (22) (23) (24) (25)
%e (111) (31) (32) (33) (34)
%e (1111) (41) (42) (43)
%e (11111) (51) (52)
%e (222) (61)
%e (111111) (124)
%e (421)
%e (1111111)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,15}]
%Y The version for differences instead of quotients is A175342.
%Y The unordered version is A342496, ranked by A342522.
%Y The strict unordered version is A342515.
%Y The distinct version is A342529.
%Y A000005 counts constant compositions.
%Y A000009 counts strictly increasing (or strictly decreasing) compositions.
%Y A000041 counts weakly increasing (or weakly decreasing) compositions.
%Y A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A167865 counts strict chains of divisors > 1 summing to n.
%Y Cf. A002843, A003242, A008965, A048004, A059966, A074206, A167606, A253249, A318991, A318992, A325557, A342528.
%K nonn
%O 0,3
%A _Gus Wiseman_, Mar 17 2021
|
