A008577 Crystal ball sequence for planar net 4.8.8.
1, 4, 9, 17, 28, 41, 57, 76, 97, 121, 148, 177, 209, 244, 281, 321, 364, 409, 457, 508, 561, 617, 676, 737, 801, 868, 937, 1009, 1084, 1161, 1241, 1324, 1409, 1497, 1588, 1681, 1777, 1876, 1977, 2081, 2188, 2297, 2409, 2524, 2641, 2761, 2884, 3009, 3137, 3268
Offset: 0
Examples
G.f. = 1 + 4*x + 9*x^2 + 17*x^3 + 28*x^4 + 41*x^5 + 67*x^6 + ... - _Michael Somos_, May 02 2020
Links
- T. D. Noe, Table of n, a(n) for n = 0..1000
- Brian Galebach, Uniform Tiling 2 of 11
- W. M. Meier and H. J. Moeck, Topology of 3-D 4-connected nets ..., J. Solid State Chem 27 1979 349-355, esp. p. 351.
- Index entries for crystal ball sequences
- Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
Programs
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Mathematica
A099837[0] = 1; A099837[n_] := Mod[n+2, 3] - Mod[n, 3]; a[n_] := 4*n/3*(n+1) + 10/9 + A099837[n+2]/9; Table[a[n], {n, 0, 39}] (* Jean-François Alcover, Feb 15 2012, after R. J. Mathar *) CoefficientList[Series[((1 + x)^2 (1 + x^2))/((1 - x)^3 (1 + x + x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 31 2015 *) LinearRecurrence[{2,-1,1,-2,1},{1,4,9,17,28},40] (* Harvey P. Dale, Dec 17 2017 *) a[ n_] := Quotient[(n^2 + n + 1)*4, 3]; (* Michael Somos, May 02 2020 *)
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PARI
a(n) = (n^2 + n + 1)*4\3; /* Michael Somos, May 02 2020 */
Formula
G.f.: ((1+x)^2*(1+x^2)) / ((1-x)^3*(1+x+x^2)). - Ralf Stephan, Apr 24 2004
a(n) = 4*(n/3)*(n+1)+10/9+A099837(n+2)/9. - R. J. Mathar, Nov 20 2010
The above g.f. and formula were originally stated as conjectures, but I now have a proof. This also justifies the b-file. Details will be added later. - N. J. A. Sloane, Dec 29 2015
From Michael Somos, May 02 2020: (Start)
Euler transform of length 3 sequence [4, -1, 1, -1].
a(n) = a(-1-n) = floor((n^2+n+1)*4/3) for all n in Z.
a(n) - 2*a(n+1) + a(n+2) = A164359(n) unless n=0.
(End)