A008530 Coordination sequence for 4-dimensional primitive di-isohexagonal orthogonal lattice.
1, 12, 60, 180, 408, 780, 1332, 2100, 3120, 4428, 6060, 8052, 10440, 13260, 16548, 20340, 24672, 29580, 35100, 41268, 48120, 55692, 64020, 73140, 83088, 93900, 105612, 118260, 131880, 146508, 162180, 178932, 196800, 215820, 236028, 257460, 280152, 304140, 329460, 356148, 384240
Offset: 0
Examples
3*a(5) = 2340 = (2*5+1)^3 + (2*5-1)^3 + (5+1)^3 + (5-1)^3. - _Bruno Berselli_, Jan 31 2013
References
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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GAP
Concatenation([1], List([1..45], n-> 6*n*(1+n^2) )); # G. C. Greubel, Nov 10 2019
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Magma
[1]cat[6*n^3+6*n: n in [1..45]]; // Vincenzo Librandi, Apr 16 2012
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Maple
1, seq( 6*k^3+6*k, k=1..45);
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Mathematica
CoefficientList[Series[(1+4*x+x^2)^2/(1-x)^4,{x,0,45}],x] (* Vincenzo Librandi, Apr 16 2012 *) LinearRecurrence[{4,-6,4,-1}, {1,12,60,180,408}, 45] (* G. C. Greubel, Nov 10 2019 *) -
PARI
vector(46, n, if(n==1,1, 6*(n-1)*(1+(n-1)^2)) ) \\ G. C. Greubel, Nov 10 2019
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Sage
[1]+[6*n*(1+n^2) for n in (1..45)] # G. C. Greubel, Nov 10 2019
Formula
G.f.: (1+4*x+x^2)^2/(1-x)^4. - Colin Barker, Apr 14 2012
3*a(n) = (2*n+1)^3 + (2*n-1)^3 + (n+1)^3 + (n-1)^3 for n>0. - Bruno Berselli, Jan 31 2013
E.g.f.: 1 + x*(12 + 18*x + 6*x^2)*exp(x). - G. C. Greubel, Nov 10 2019
Comments