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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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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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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REFERENCES
| I. J. Good, Problem 6137, American Mathematical Monthly 1978 pages 830-831.
Hansraj Gupta, Finite Differences of the Partition Function, pp. 1241-1243.
A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), pp. 237-254.
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LINKS
| Almkvist, Gert, "On the differences of the partition function", Acta Arith., 61.2 (1992), 173-181.
Weisstein, Eric W. "Forward Difference".
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FORMULA
| Odlyzko gives an asymptotic formula a(n)~(6/(Pi)^2) * (n log n)^2
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MAPLE
| 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); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jul 20 2009]
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CROSSREFS
| Cf. A000041, A002865, A053445, A072380, A081094, A081095, A175804, A155861.
Sequence in context: A189713 A162267 A009017 * A005745 A045618 A038737
Adjacent sequences: A119709 A119710 A119711 * A119713 A119714 A119715
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KEYWORD
| nonn
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AUTHOR
| Moshe Newman (moshnoiman(AT)gmail.com), Jun 11 2006
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EXTENSIONS
| a(11) - a(41) from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jul 20 2009
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