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 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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved