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A155861
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a(n) is the smallest integer k such that the n-th (backward) difference of the partition sequence A000041 is positive from k onwards.
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2
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1, 2, 8, 26, 68, 134, 228, 352, 510, 704, 934, 1204, 1514, 1866, 2260, 2702, 3188, 3722, 4304, 4936, 5620, 6354, 7140, 7980, 8872, 9822, 10826, 11888, 13006, 14182, 15416, 16712, 18066, 19480, 20956, 22494, 24096, 25760, 27486, 29278, 31134
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OFFSET
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0,2
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COMMENTS
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Using a different (forward) definition of the difference operator, this sequence has also been given as 0,1,6,23,64,129,222,... A119712.
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LINKS
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Table of n, a(n) for n=0..40.
Almkvist, Gert, "On the differences of the partition function", Acta Arith., 61.2 (1992), 173-181.
Knessl, Charles, "Asymptotic Behavior of High-Order Differences of the Partition Function", Communications on Pure and Applied Mathematics, 44 (1991), 1033-1045.
Odlyzko, A. M., "Differences of the partition function", Acta Arith., 49 (1988), 237-254.
Weisstein, Eric W., "Backward Difference".
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FORMULA
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An asymptotic formula is a(n) ~ 6/Pi^2 * n^2 (log n)^2.
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MAPLE
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A41:= n-> `if` (n<0, 0, combinat[numbpart](n)):
DB:= proc(p)
proc(n) option remember;
p(n) -p(n-1)
end
end:
a:= proc(n) option remember;
local f, k;
if n=0 then 1
else f:= (DB@@n)(A41);
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);
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CROSSREFS
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Cf. A000041, A002865, A053445, A072380, A081094, A081095, A175804, A119712.
Sequence in context: A099416 A211885 A101696 * A212140 A136594 A167826
Adjacent sequences: A155858 A155859 A155860 * A155862 A155863 A155864
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, Dec 16 2010
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STATUS
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approved
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