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Search: id:A001082
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| A001082 |
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a(n) = n(3n-4)/4 if n even, (n-1)(3n+1)/4 if n odd. |
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+0
27
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| 0, 1, 5, 8, 16, 21, 33, 40, 56, 65, 85, 96, 120, 133, 161, 176, 208, 225, 261, 280, 320, 341, 385, 408, 456, 481, 533, 560, 616, 645, 705, 736, 800, 833, 901, 936, 1008, 1045, 1121, 1160, 1240, 1281, 1365, 1408, 1496, 1541, 1633, 1680, 1776, 1825, 1925, 1976
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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3*a(n)+1 is a perfect square.
Could also be called generalized octagonal numbers, or n(3n-2) for n=0, +- 1, +- 2,.... Cf. A001318, generalized pentagonal numbers. - Matthew Vandermast (ghodges14(AT)comcast.net), Apr 10 2003
n^2 - n - floor(n/2)^2.
Sequence allows us to find X values of the equation: 3*X^3 + X^2 = Y^2. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007
Number of units of a(n) belongs to a periodic sequence: 0, 1, 5, 8, 6, 1, 3, 0, 6, 5, 5, 6, 0, 3, 1, 6, 8, 5, 1, 0. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]
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LINKS
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Index entries for sequences related to linear recurrences with constant coefficients
R. Stephan, On the solutions to 'px+1 is square'
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FORMULA
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G.f.: sum_{n=0..inf} (-1)^n*[x^(a(2n+1)) + x^(a(2n+2))] = 1/1 - (x-x^2)/1 - (x^2-x^4)/1 - (x^3-x^6)/1 -...- (x^k - x^(2k))/1 -... (continued fraction where k=1..inf). - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 16 2002
a(2n)=n(3n+2), a(2n+1)=3*n^2+4n+1. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007
a(n+1) = ceil(n/2)^2+A046092([n/2]).
a(2n)=n(3n-2)=A000567(n), a(2n+1)=n(3n+2)=A045944(n). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007
O.g.f.: -x^2*(x^2+4*x+1)/((x-1)^3*(1+x)^2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 15 2008
a(n+1)-a(n)=A022998(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 15 2008
a(n) = n^2+n-ceiling(n/2)^2 with offset 0..a(0)=0 [From Gary Detlefs (gdetlefs(AT)aol.com), Feb 23 2010]
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MAPLE
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seq(n^2+n-ceil(n/2)^2, n=0..51); [From Gary Detlefs (gdetlefs(AT)aol.com), Feb 23 2010]
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MATHEMATICA
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f[n_]:=IntegerQ[Sqrt[1+3*n]]; Select[Range[0, 8! ], f[ # ]&] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 19 2010]
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PROGRAM
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(PARI) a(n)=if(n%2, (n-1)*(3*n+1)/4, n*(3*n-4)/4)
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CROSSREFS
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Partial sums of A022998.
Cf. A005563, A046092.
Sequence in context: A141536 A065905 A126695 this_sequence A030006 A088586 A073136
Adjacent sequences: A001079 A001080 A001081 this_sequence A001083 A001084 A001085
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KEYWORD
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nonn,easy,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com) and Tom Duff
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000
More terms from Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 25 2003
Some of the formulae were corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 15 2008
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