A345897 a(n) = 2*n^4/3 - 4*n^3/3 + 11*n^2/6 - 13*n/6 + 1.

1, 0, 4, 29, 107, 286, 630, 1219, 2149, 3532, 5496, 8185, 11759, 16394, 22282, 29631, 38665, 49624, 62764, 78357, 96691, 118070, 142814, 171259, 203757, 240676, 282400, 329329, 381879, 440482, 505586, 577655, 657169, 744624, 840532, 945421, 1059835, 1184334, 1319494

Offset

0,3

Comments

For n >=1, a(n) is the number of divisions of a 2 X n board into 3 pieces. See Jacob Brown article.

Links

Michael De Vlieger, 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.

Tomislav Došlić and Luka Podrug, Sweet division problems: from chocolate bars to honeycomb strips and back, arXiv:2304.12121 [math.CO], 2023.

Samuel Durham and Tom Richmond, Connected Subsets of an n X 2 Rectangle, The College Mathematics Journal, Volume 51, 2020 - Issue 1.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

From Chai Wah Wu, Jun 29 2021: (Start)

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.

G.f.: (1 - 5*x + 14*x^2 - x^3 + 7*x^4)/(1-x)^5. (End)

Mathematica

CoefficientList[Series[(1 - 5 x + 14 x^2 - x^3 + 7 x^4)/(1 - x)^5, {x, 0, 38}], x] (* Michael De Vlieger, Apr 28 2023 *)

Prog

(PARI) a(n) = 2*n^4/3 - 4*n^3/3 + 11*n^2/6 - 13*n/6 + 1;

Crossrefs

Cf. A172482 (same but where the rightmost squares separate).

Sequence in context: A199399 A211545 A295842 * A368372 A095670 A370435

Adjacent sequences: A345894 A345895 A345896 * A345898 A345899 A345900

Keyword

nonn,easy

Author

Michel Marcus, Jun 29 2021

Status

approved