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%I #39 Jan 28 2022 03:20:18
%S 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,
%T 178,204,239,271,316,361,416,472,545,618,706,800,912,1032,1173,1320,
%U 1496,1687,1902,2137,2410,2702,3034,3398,3808,4258,4765,5313,5932,6613
%N Number of partitions of n such that the distinct terms arranged in increasing order form a string of two or more consecutive integers.
%C Number of partitions of n with maximal distance between parts = 1; column k=1 of A238353. [_Joerg Arndt_, Mar 23 2014]
%C 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
%H Alois P. Heinz, <a href="/A237665/b237665.txt">Table of n, a(n) for n = 0..1000</a>
%H Shane Chern (Xiaohang Chen), <a href="https://sites.psu.edu/shanechern/files/2018/07/On-a-conjecture-of-George-Beck-II-2dpatgk.pdf">On a conjecture of George Beck. II</a>, 2018.
%F a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Jan 28 2022
%e 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.)
%p b:= proc(n, i, t) option remember;
%p `if`(n=0 or i=1, `if`(n=0 and t=2 or n>0 and t>0, 1, 0),
%p `if`(i>n, 0, add(b(n-i*j, i-1, min(t+1, 2)), j=1..n/i)))
%p end:
%p a:= n-> add(b(n, i, 0), i=1..n):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Feb 15 2014
%t Map[Length[Select[Map[Differences[DeleteDuplicates[#]] &, IntegerPartitions[#]], (Table[-1, {Length[#]}] == # && # =!= \{}) &]] &, Range[55]] (* _Peter J. C. Moses_, Feb 09 2014 *)
%t 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_ *)
%Y Cf. A034296, A237666, A092265.
%K nonn,easy
%O 0,6
%A _Clark Kimberling_, Feb 11 2014
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