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 A237665 Number of partitions of n such that the distinct terms arranged in increasing order form a string of two or more consecutive integers. 6
 0, 0, 0, 1, 1, 3, 3, 6, 6, 10, 11, 16, 17, 24, 27, 35, 39, 50, 57, 70, 79, 97, 111, 132, 150, 178, 204, 239, 271, 316, 361, 416, 472, 545, 618, 706, 800, 912, 1032, 1173, 1320, 1496, 1687, 1902, 2137, 2410, 2702, 3034, 3398, 3808, 4258, 4765, 5313, 5932, 6613 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Number of partitions of n with maximal distance between parts = 1; column k=1 of A238353. [Joerg Arndt, Mar 23 2014] Conjecture:  a(n+1) = sum of smallest parts in the distinct partitions of n with an even number of parts. - George Beck, May 06 2017 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 Shane Chern (Xiaohang Chen), On a conjecture of George Beck. II, 2018. EXAMPLE The qualifying partitions of 8 are 332, 3221, 32111, 22211, 221111, 2111111, so that a(8) = 6.  (The strings of distinct parts arranged in increasing order are 23, 123, 123, 12, 12, 12.) MAPLE b:= proc(n, i, t) option remember;       `if`(n=0 or i=1, `if`(n=0 and t=2 or n>0 and t>0, 1, 0),       `if`(i>n, 0, add(b(n-i*j, i-1, min(t+1, 2)), j=1..n/i)))     end: a:= n-> add(b(n, i, 0), i=1..n): seq(a(n), n=0..60);  # Alois P. Heinz, Feb 15 2014 MATHEMATICA Map[Length[Select[Map[Differences[DeleteDuplicates[#]] &, IntegerPartitions[#]], (Table[-1, {Length[#]}] == # && # =!= \{}) &]] &, Range[55]] (* Peter J. C. Moses, Feb 09 2014 *) b[n_, i_, t_] := b[n, i, t] = If[n==0 || i==1, If[n==0 && t==2 || n>0 && t > 0, 1, 0], If[i>n, 0, Sum[b[n-i*j, i-1, Min[t+1, 2]], {j, 1, n/i}]]]; a[n_] := Sum[b[n, i, 0], {i, 1, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 17 2015, after Alois P. Heinz *) CROSSREFS Cf. A034296, A237666, A092265. Sequence in context: A008805 A188270 A026925 * A088528 A220153 A219627 Adjacent sequences:  A237662 A237663 A237664 * A237666 A237667 A237668 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 11 2014 STATUS approved

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Last modified October 14 12:45 EDT 2019. Contains 328006 sequences. (Running on oeis4.)