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 A342493 Number of compositions of n with strictly increasing first quotients. 6
 1, 1, 2, 3, 6, 8, 11, 16, 22, 28, 39, 49, 61, 77, 93, 114, 140, 169, 198, 233, 276, 321, 381, 439, 509, 591, 678, 774, 883, 1007, 1147, 1300, 1465, 1641, 1845, 2068, 2317, 2590, 2881, 3193, 3549, 3928, 4341, 4793, 5282, 5813, 6401, 7027, 7699, 8432, 9221, 10076 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..500 Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence. EXAMPLE The composition (3,1,1,2) has first quotients (1/3,1,2) so is counted under a(7). The a(1) = 1 through a(7) = 16 compositions:   (1)  (2)    (3)    (4)      (5)      (6)        (7)        (1,1)  (1,2)  (1,3)    (1,4)    (1,5)      (1,6)               (2,1)  (2,2)    (2,3)    (2,4)      (2,5)                      (3,1)    (3,2)    (3,3)      (3,4)                      (1,1,2)  (4,1)    (4,2)      (4,3)                      (2,1,1)  (1,1,3)  (5,1)      (5,2)                               (2,1,2)  (1,1,4)    (6,1)                               (3,1,1)  (2,1,3)    (1,1,5)                                        (3,1,2)    (2,1,4)                                        (4,1,1)    (2,2,3)                                        (2,1,1,2)  (3,1,3)                                                   (3,2,2)                                                   (4,1,2)                                                   (5,1,1)                                                   (2,1,1,3)                                                   (3,1,1,2) MAPLE b:= proc(n, q, l) option remember; `if`(n=0, 1, add(      `if`(q=0 or q>l/j, b(n-j, l/j, j), 0), j=1..n))     end: a:= n-> b(n, 0\$2): seq(a(n), n=0..55);  # Alois P. Heinz, Mar 25 2021 MATHEMATICA Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@Divide@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 15}] (* Second program: *) b[n_, q_, l_] := b[n, q, l] = If[n == 0, 1, Sum[      If[q == 0 || q > l/j, b[n - j, l/j, j], 0], {j, 1, n}]]; a[n_] := b[n, 0, 0]; a /@ Range[0, 55] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *) CROSSREFS The version for differences instead of quotients is A325547. The weakly increasing version is A342492. The strictly decreasing version is A342494. The unordered version is A342498, ranked by A342524. The strict unordered version is A342517. A000005 counts constant compositions. A000009 counts strictly increasing (or strictly decreasing) compositions. A000041 counts weakly increasing (or weakly decreasing) compositions. A001055 counts factorizations. A003238 counts chains of divisors summing to n - 1 (strict: A122651). A074206 counts ordered factorizations. A167865 counts strict chains of divisors > 1 summing to n. A274199 counts compositions with all adjacent parts x < 2y. Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A318991, A318992, A342527, A342528. Sequence in context: A185599 A211519 A329384 * A325547 A242340 A033766 Adjacent sequences:  A342490 A342491 A342492 * A342494 A342495 A342496 KEYWORD nonn AUTHOR Gus Wiseman, Mar 16 2021 EXTENSIONS a(21)-a(51) from Alois P. Heinz, Mar 18 2021 STATUS approved

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Last modified October 24 08:15 EDT 2021. Contains 348217 sequences. (Running on oeis4.)