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A002843
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Number of partitions of n into parts 1/2, 3/4, 7/8, etc.
(Formerly M1072 N0405)
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1
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1, 1, 2, 4, 7, 13, 24, 43, 78, 141, 253, 456, 820, 1472, 2645, 4749, 8523, 15299, 27456, 49267, 88407, 158630, 284622, 510683, 916271, 1643963, 2949570, 5292027, 9494758, 17035112, 30563634, 54835835, 98383803, 176515310, 316694823
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| The g.f. (z**2+z+1)*(z-1)**2/(1-2*z-z**3+3*z**4) conjectured by S. Plouffe in his 1992 dissertation is wrong.
Also number of compositions (a_1,a_2,...) of n with each a_i <= 2*a_(i-1). [From Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 02 2009]
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REFERENCES
| Minc, H.; A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid. Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| Row sums of A049286 and A047913. [From Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 02 2009]
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EXAMPLE
| A straightforward partition problem: 1=1/2 + 1/2 and there is no other partition of 1, so a(1)=1.
a(3)=4 since 3 = 6(1/2) = 4(3/4) = 2(3/4)+3(1/2) = 2(7/8)+3/4+1/2.
a(4)=7 since 4 = 8(1/2) = 5(1/2)+2(3/4) = 2(1/2)+4(3/4) = 3(1/2)+3/4+2(7/8) = 3(3/4)+2(7/8) = 1/2+4(7/8) = 2(15/16)+7/8+3/4+1/2.
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CROSSREFS
| Cf. A047913.
Sequence in context: A006745 A049284 A049285 * A128742 A107281 A006744
Adjacent sequences: A002840 A002841 A002842 * A002844 A002845 A002846
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from John W. Layman (layman(AT)math.vt.edu), Nov 24 2001
Examples and offset corrected by Larry Reeves (larryr(AT)acm.org), Jan 06 2005
Further terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 13 2006
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