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A057566
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Number of collinear triples in a 3 X n rectangular grid.
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0
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0, 1, 2, 8, 20, 43, 78, 130, 200, 293, 410, 556, 732, 943, 1190, 1478, 1808, 2185, 2610, 3088, 3620, 4211, 4862, 5578, 6360, 7213, 8138, 9140, 10220, 11383, 12630, 13966, 15392, 16913, 18530, 20248, 22068, 23995, 26030, 28178, 30440, 32821, 35322
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (3,-2,-2,3,-1). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 23 2010]
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FORMULA
| Conjecture: a(n)=5Floor[(2n^3-3n^2-n)/24]+Floor[(2(n-1)^3-3(n-1)^2-(n-1))/24]+n, which fits all of the listed terms.
a(n)=a(n-1)+b(n), with a(0)=-2, b(0)=8 and being b(n)=b(n-1)-7+Sum_{k=0..n}{5*(k mod 2)+[(k+1) mod 2]} - Paolo P. Lava (paoloplava(AT)gmail.com), Aug 24 2007
a(n)= +3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5) = n^3/2-n^2+n+(1-(-1)^n)/4. G.f.: x*(1-x+4*x^2+2*x^3)/((1+x)*(x-1)^4). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 23 2010]
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MAPLE
| P:=proc(n) local a, b, i; b:=8; a:=-2; for i from 0 by 1 to n do b:=b-7+sum('(5*(k mod 2)+((k+1) mod 2))', 'k'=0..i); a:=a+b; print(a); od; end: P(200); - Paolo P. Lava (paoloplava(AT)gmail.com), Aug 24 2007
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CROSSREFS
| Second differences give A047264. Third differences are periodic {5, 1, 5, 1, ...} and form A010686. See A000938 for the n X n grid.
Sequence in context: A203420 A048096 A072250 * A009303 A096586 A165751
Adjacent sequences: A057563 A057564 A057565 * A057567 A057568 A057569
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KEYWORD
| nonn
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AUTHOR
| John W. Layman (layman(AT)math.vt.edu), Oct 04 2000
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