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A237666 Number of partitions of n that include a pair of consecutive integers.

%I A237666 #21 Jan 28 2022 07:06:36
%S A237666 0,0,0,1,1,3,3,7,9,15,20,32,40,61,78,112,142,199,250,341,428,568,710,
%T A237666 930,1151,1486,1835,2334,2868,3615,4413,5513,6706,8298,10052,12359,
%U A237666 14895,18195,21857,26526,31747,38337,45702,54923,65272,78062,92481,110168,130089
%N A237666 Number of partitions of n that include a pair of consecutive integers.
%H A237666 Vaclav Kotesovec, <a href="/A237666/b237666.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A237666 a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). - _Vaclav Kotesovec_, Jan 28 2022
%F A237666 Conjecture: for n > 0, a(n) = A000041(n) - A116931(n). - _Vaclav Kotesovec_, Jan 28 2022
%e A237666 The qualifying partitions of 8 are 521, 431, 332, 421, 3221, 32111, 22211, 221111, 2111111, so that a(8) = 9.
%p A237666 g:= proc(n, i) option remember; `if`(n=0, 1,
%p A237666       `if`(i<1, 0, add(g(n-i*j, i-1), j=0..n/i)))
%p A237666     end:
%p A237666 b:= proc(n, i, l) option remember; `if`(n=0 or i<1, 0,
%p A237666        b(n, i-1, 0) +add(`if`(i+1=l, g(n-i*j, i-1),
%p A237666        b(n-i*j, i-1, i)), j=1..n/i))
%p A237666     end:
%p A237666 a:= n-> b(n$2, 0):
%p A237666 seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 14 2014
%t A237666 Map[Length[Cases[Map[Differences[DeleteDuplicates[#]] &, IntegerPartitions[#]], {___, -1, ___}]] &, Range[50]]  (* _Peter J. C. Moses_, Feb 09 2014 *)
%t A237666 g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Sum[g[n-i*j, i-1], {j, 0, n/i}]]]; b[n_, i_, l_] := b[n, i, l] = If[n==0 || i<1, 0, b[n, i-1, 0] + Sum[If[i+1 == l, g[n-i*j, i-1], b[n-i*j, i-1, i]], {j, 1, n/i}]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Sep 01 2016, after _Alois P. Heinz_ *)
%Y A237666 Cf. A116931, A237665.
%K A237666 nonn,easy
%O A237666 0,6
%A A237666 _Clark Kimberling_, Feb 11 2014