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A119712
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a(n) is the smallest integer k such that the n-th (forward) difference of the partition sequence A000041 is positive from k onwards.
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3
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0, 1, 6, 23, 64, 129, 222, 345, 502, 695, 924, 1193, 1502, 1853, 2246, 2687, 3172, 3705, 4286, 4917, 5600, 6333, 7118, 7957, 8848, 9797, 10800, 11861, 12978, 14153, 15386, 16681, 18034, 19447, 20922, 22459, 24060, 25723, 27448, 29239, 31094, 33015
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OFFSET
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0,3
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COMMENTS
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The first entry is considered to be indexed by zero. For example, the third difference A072380 starts with -1,1 and continues alternating in sign till the 24th entry, from which point it is positive.
Using a different (backward) definition of the difference operator, this sequence has also been given as 1, 8, 26, 68, 134, 228, 352, ... A155861.
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LINKS
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I. J. Good, Problem 6137, American Mathematical Monthly, 1978, pages 830-831.
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FORMULA
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Odlyzko gives an asymptotic formula a(n)~(6/(Pi)^2) * (n log n)^2
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MAPLE
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with(combinat): DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end: a:= proc(n) option remember; local f, k; if n=0 then 0 else f:= (DD@@n)(numbpart); for k from a(n-1) while not (f(k)>0 and f(k+1)>0) do od; k fi end: seq(a(n), n=0..20); # Alois P. Heinz, Jul 20 2009
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MATHEMATICA
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a[n_] := a[n] = Module[{f}, f[i_] = DifferenceDelta[PartitionsP[i], {i, n}]; For[j = 2, True, j++, If[f[j] > 0 && f[j+1] > 0, Return[j]]]];
a[0] = 0; a[1] = 1;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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