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A237666
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Number of partitions of n that include a pair of consecutive integers.
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3
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0, 0, 0, 1, 1, 3, 3, 7, 9, 15, 20, 32, 40, 61, 78, 112, 142, 199, 250, 341, 428, 568, 710, 930, 1151, 1486, 1835, 2334, 2868, 3615, 4413, 5513, 6706, 8298, 10052, 12359, 14895, 18195, 21857, 26526, 31747, 38337, 45702, 54923, 65272, 78062, 92481, 110168, 130089
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OFFSET
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0,6
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LINKS
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FORMULA
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EXAMPLE
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The qualifying partitions of 8 are 521, 431, 332, 421, 3221, 32111, 22211, 221111, 2111111, so that a(8) = 9.
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MAPLE
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g:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(g(n-i*j, i-1), j=0..n/i)))
end:
b:= proc(n, i, l) option remember; `if`(n=0 or i<1, 0,
b(n, i-1, 0) +add(`if`(i+1=l, g(n-i*j, i-1),
b(n-i*j, i-1, i)), j=1..n/i))
end:
a:= n-> b(n$2, 0):
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MATHEMATICA
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Map[Length[Cases[Map[Differences[DeleteDuplicates[#]] &, IntegerPartitions[#]], {___, -1, ___}]] &, Range[50]] (* Peter J. C. Moses, Feb 09 2014 *)
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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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