A172482 a(n) = (1+n)*(9 + 11*n + 4*n^2)/3.

3, 16, 47, 104, 195, 328, 511, 752, 1059, 1440, 1903, 2456, 3107, 3864, 4735, 5728, 6851, 8112, 9519, 11080, 12803, 14696, 16767, 19024, 21475, 24128, 26991, 30072, 33379, 36920, 40703, 44736, 49027, 53584, 58415, 63528, 68931, 74632, 80639, 86960, 93603

Offset: 0

Author

Paul Curtz, Feb 04 2010

Keywords

• nonn
• easy

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×n Rectangular Grid, arXiv:2106.14755 [math.CO], 2021.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A100178, A131941.

Programs

Magma

[(1+n)*(9+11*n+4*n^2)/3: n in [0..40]]; // Vincenzo Librandi, Aug 04 2011

Mathematica

CoefficientList[Series[(x + 3) (1 + x)/(x - 1)^4, {x, 0, 40}], x] (* Michael De Vlieger, Nov 02 2018 *)

PARI

a(n)=(1+n)*(9+11*n+4*n^2)/3 \\ Charles R Greathouse IV, Aug 04 2011

Formula

a(n) = A131941(2n+2), where A100178(n) = A131941(2n-1).

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

Extensions

Edited by R. J. Mathar, Feb 24 2010