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

0,2

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).

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)

Example

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

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 *)

Prog

(PARI) {a(n) = (n^2 + n + 1)*4\3)}; /* Michael Somos, May 02 2020 */

Crossrefs

Partial sums of A008576.

Cf. A099837, A164359.

Sequence in context: A033617 A033613 A033608 * A033605 A301018 A008137

Adjacent sequences: A008574 A008575 A008576 * A008578 A008579 A008580

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved