OFFSET
0,1
COMMENTS
One of the bisections of the left central column in the Janet table A172002.
Row 1 of the convolution array A213844. - Clark Kimberling, Jul 05 2012
With offset 2, this is 4*n^3/3 - 3*n^2 + 8*n/3 - 1, the number of divisions of a 2 X n board into 3 pieces where the rightmost squares separate. See Jacob Brown article. - Michel Marcus, Jun 29 2021
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Jacob Brown, Counting Divisions of a 2 X n Rectangular Grid, arXiv:2106.14755 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = 4*a(n) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) mod 10 = 3, 6, 7, 4, 5, 8, 1, 2, 9, 0 (and repeat periodically).
G.f.: (x+3)*(1+x)/(x-1)^4.
E.g.f.: exp(x)*(9 + 39*x + 27*x^2 + 4*x^3)/3. - Stefano Spezia, Mar 02 2025
a(n) = Sum_{i=0..n} (2*i + 1)*(4*n - 4*i + 3). - Derek Delk, Jan 17 2026
MATHEMATICA
CoefficientList[Series[(x + 3) (1 + x)/(x - 1)^4, {x, 0, 40}], x] (* Michael De Vlieger, Nov 02 2018 *)
PROG
(Magma) [(1+n)*(9+11*n+4*n^2)/3: n in [0..40]]; // Vincenzo Librandi, Aug 04 2011
(PARI) a(n)=(1+n)*(9+11*n+4*n^2)/3 \\ Charles R Greathouse IV, Aug 04 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Feb 04 2010
EXTENSIONS
Edited by R. J. Mathar, Feb 24 2010
STATUS
approved