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A212964
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Number of (w,x,y) with all terms in {0,...,n} and |w-x| < |x-y| < |y-w|.
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11
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0, 0, 0, 2, 6, 14, 26, 44, 68, 100, 140, 190, 250, 322, 406, 504, 616, 744, 888, 1050, 1230, 1430, 1650, 1892, 2156, 2444, 2756, 3094, 3458, 3850, 4270, 4720, 5200, 5712, 6256, 6834, 7446, 8094, 8778, 9500, 10260, 11060, 11900, 12782, 13706
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OFFSET
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0,4
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COMMENTS
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For a guide to related sequences, see A212959.
Magic numbers of nucleons in a biaxially deformed nucleus at oscillator ratio 1:2 (oblate ellipsoid) under the simple harmonic oscillator model. - Jess Tauber, May 14 2013
a(n) is the number of Sidon subsets of {1,...,n+1} of size 3. - Carl Najafi, Apr 27 2014
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LINKS
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FORMULA
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a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5).
G.f.: f(x)/g(x), where f(x)=2*x^3 and g(x)=(1+x)(1-x)^4.
a(n) = Sum_{i=1..n} floor(i^2/2) - 2*floor(i/2). - Wesley Ivan Hurt, Jul 23 2014
a(n) = (2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8. - Robert Israel, Jul 23 2014
E.g.f.: (x*(2*x^2 + 3*x - 3)*cosh(x) + (2*x^3 + 3*x^2 - 3*x + 3)*sinh(x))/12. - Stefano Spezia, Jul 06 2021
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MAPLE
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MATHEMATICA
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t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] < Abs[x - y] < Abs[y - w], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]] (* A212964 *)
CoefficientList[Series[2 x^3/((1 + x) (1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 25 2014 *)
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PROG
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(Magma) [(2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8: n in [0..50]]; // Vincenzo Librandi, Jul 25 2014
(PARI) a(n) = (2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8;
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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