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A001971
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Nearest integer to n^2/8.
(Formerly M0625 N0227)
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6
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0, 0, 1, 1, 2, 3, 5, 6, 8, 10, 13, 15, 18, 21, 25, 28, 32, 36, 41, 45, 50, 55, 61, 66, 72, 78, 85, 91, 98, 105, 113, 120, 128, 136, 145, 153, 162, 171, 181, 190, 200, 210, 221, 231, 242, 253, 265, 276, 288, 300, 313, 325, 338, 351, 365, 378, 392, 406, 421, 435, 450
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Restricted partitions.
Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.de), Jun 05 2009: (Start)
a(0,..,2)=0; a(n) are the partitions of floor((3*n+3)/2) with 3 distinct
numbers of the set {1,..,n}; partitions of floor((3*n+3)/2)-C and
ceiling((3*n+3)/2)+C have equal numbers. (End)
Odd terms are the triangular numbers, even terms are the midpoint (rounded up where necessary) of the surrounding odd terms. [From Carl R. White (oeisfan(AT)phodd.net), Aug 12 2010]
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REFERENCES
| G. Almkvist, Invariants, mostly old ones. Pacific J. Math. 86 (1980), no. 1, 1-13. MR0586866 (81j:14029)
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 276-281.
M. Jeger,Einfuehrung in die Kombinatorik,Klett,1975,Bd.2,pages 110- [From Weisenhorn Paul (paulweisenhorn(AT)online.de), Jun 05 2009]
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
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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.
Index to sequences with linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).
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FORMULA
| The listed terms through a(18)=50 satisfy a(n)=a(n-4)+n - John W. Layman (layman(AT)math.vt.edu), Dec 16 1999
G.f.: x^2(1-x+x^2)/(1-2x+x^2-x^4+2x^5-x^6)=x^2(1-x^6)/((1-x)(1-x^2)(1-x^3)(1-x^4)). - Michael Somos Feb 07 2004
Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.de), Jun 05 2009: (Start)
a(n)=floor((n^2-2*n+5)/8); GF: product[s=1..3] (x^s-x^(n+1))/(1-x^s);
(End)
Contribution from Carl R. White (oeisfan(AT)phodd.net), Aug 12 2010: (Start)
a(2*n+1) = A000217(n)
a(2*n)=floor( (A000217(n-1)+A000217(n)+1)/2 ) (End)
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MAPLE
| A001971:=-(1-z+z**2)/((z+1)*(z**2+1)*(z-1)**3); [Conjectured (correctly) by S. Plouffe in his 1992 dissertation.]
Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.de), Jun 05 2009: (Start)
a(n):=subs({x=1}, convert(series(product((1-x^i), i=n-2..n)/((1-x^2)*(1-x^3)),
x, floor((3*n+5)/2)), polynom)); (End)
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PROG
| (PARI) a(n)=round(n^2/8)
(MAGMA) [Round(n^2/8): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011
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CROSSREFS
| The 4th diagonal of A061857?
Cf. A000217 [From Carl R. White (oeisfan(AT)phodd.net), Aug 12 2010]
Sequence in context: A049862 A022829 A056837 * A122493 A053873 A118053
Adjacent sequences: A001968 A001969 A001970 * A001972 A001973 A001974
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited Feb 08 2004
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