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 A182699 Number of emergent parts in all partitions of n. 25
 0, 0, 0, 0, 1, 1, 4, 4, 10, 12, 22, 27, 47, 56, 89, 112, 164, 205, 294, 364, 505, 630, 845, 1052, 1393, 1719, 2235, 2762, 3533, 4343, 5506, 6730, 8443, 10296, 12786, 15531, 19161, 23161, 28374, 34201, 41621, 49975, 60513, 72385, 87200, 103999, 124670, 148209 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS Here the "emergent parts" of the partitions of n are defined to be the parts (with multiplicity) of all the partitions that do not contain "1" as a part, removed by one copy of the smallest part of every partition. Note that these parts are located in the head of the last section of the set of partitions of n. Also, here the "filler parts" of the partitions of n are defined to be the parts of the last section of the set of partitions of n that are not the emergent parts. For n >= 4, length of row n of A183152. - Omar E. Pol, Aug 08 2011 Also total number of parts of the regions that do not contain 1 as a part in the last section of the set of partitions of n (cf. A083751, A187219). - Omar E. Pol, Mar 04 2012 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA a(n) = A138135(n) - A002865(n), n >= 1. From Omar E. Pol, Oct 21 2011: (Start) a(n) = A006128(n) - A006128(n-1) - A000041(n), n >= 1. a(n) = A138137(n) - A000041(n), n >= 1. (End) a(n) = A076276(n) - A006128(n-1), n >= 1. - Omar E. Pol, Oct 30 2011 EXAMPLE For n = 6 the partitions of 6 contain four "emergent" parts: (3), (4), (2), (2), so a(6) = 4. See below the location of the emergent parts. 6 (3) + 3 (4) + 2 (2) + (2) + 2 5 + 1 3 + 2 + 1 4 + 1 + 1 2 + 2 + 1 + 1 3 + 1 + 1 + 1 2 + 1 + 1 + 1 + 1 1 + 1 + 1 + 1 + 1 + 1 For a(10) = 22 see the link for the location of the 22 "emergent parts" (colored yellow and green) and the location of the 42 "filler parts" (colored blue) in the last section of the set of partitions of 10. MAPLE b:= proc(n, i) option remember; local t, h;       if n<0 then [0, 0, 0]     elif n=0 then [0, 1, 0]     elif i<2 then [0, 0, 0]     else t:= b(n, i-1); h:= b(n-i, i);          [t[1]+h[1]+h[2], t[2], t[3]+h[3]+h[1]]       fi     end: a:= n-> b(n, n)[3]: seq (a(n), n=0..50);  # Alois P. Heinz, Oct 21 2011 MATHEMATICA b[n_, i_] := b[n, i] = Module[{t, h}, Which[n<0, {0, 0, 0}, n == 0, {0, 1, 0}, i<2 , {0, 0, 0}, True, t = b[n, i-1]; h = b[n-i, i]; Join [t[[1]] + h[[1]] + h[[2]], t[[2]], t[[3]] + h[[3]] + h[[1]] ]]]; a[n_] := b[n, n][[3]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 18 2015, after Alois P. Heinz *) CROSSREFS Cf. A000041, A002865, A135010, A138121, A138135, A182700, A182709, A182740. Sequence in context: A006477 A233739 A279036 * A058596 A180964 A237668 Adjacent sequences:  A182696 A182697 A182698 * A182700 A182701 A182702 KEYWORD nonn,easy AUTHOR Omar E. Pol, Nov 29 2010 STATUS approved

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Last modified June 19 11:14 EDT 2019. Contains 324219 sequences. (Running on oeis4.)