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 A342494 #20 Feb 16 2025 08:34:01
%S A342494 1,1,2,3,5,8,12,15,21,30,39,50,65,82,103,129,160,196,240,293,352,422,
%T A342494 500,593,706,832,974,1138,1324,1534,1783,2054,2362,2712,3108,3552,
%U A342494 4051,4606,5232,5935,6713,7573,8536,9597,10773,12085,13534,15119,16874,18809
%N A342494 Number of compositions of n with strictly decreasing first quotients.
%C A342494 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 A342494 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342494 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A342494 The composition (1,2,3,4,2) has first quotients (2,3/2,4/3,1/2) so is counted under a(12).
%e A342494 The a(1) = 1 through a(6) = 12 compositions:
%e A342494 (1) (2) (3) (4) (5) (6)
%e A342494 (1,1) (1,2) (1,3) (1,4) (1,5)
%e A342494 (2,1) (2,2) (2,3) (2,4)
%e A342494 (3,1) (3,2) (3,3)
%e A342494 (1,2,1) (4,1) (4,2)
%e A342494 (1,2,2) (5,1)
%e A342494 (1,3,1) (1,2,3)
%e A342494 (2,2,1) (1,3,2)
%e A342494 (1,4,1)
%e A342494 (2,3,1)
%e A342494 (3,2,1)
%e A342494 (1,2,2,1)
%t A342494 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]
%Y A342494 The weakly decreasing version is A069916.
%Y A342494 The version for differences instead of quotients is A325548.
%Y A342494 The strictly increasing version is A342493.
%Y A342494 The unordered version is A342499, ranked by A342525.
%Y A342494 The strict unordered version is A342518.
%Y A342494 A000005 counts constant compositions.
%Y A342494 A000009 counts strictly increasing (or strictly decreasing) compositions.
%Y A342494 A000041 counts weakly increasing (or weakly decreasing) compositions.
%Y A342494 A001055 counts factorizations.
%Y A342494 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342494 A074206 counts ordered factorizations.
%Y A342494 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342494 A274199 counts compositions with all adjacent parts x < 2y.
%Y A342494 Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A318991, A318992, A342527, A342528.
%K A342494 nonn
%O A342494 0,3
%A A342494 _Gus Wiseman_, Mar 17 2021
%E A342494 a(21)-a(49) from _Alois P. Heinz_, Mar 18 2021