A273895 -id:A273895 - cp's OEIS frontend

A001169 Number of board-pile polyominoes with n cells.

1, 2, 6, 19, 61, 196, 629, 2017, 6466, 20727, 66441, 212980, 682721, 2188509, 7015418, 22488411, 72088165, 231083620, 740754589, 2374540265, 7611753682, 24400004911, 78215909841, 250726529556, 803721298537, 2576384425157, 8258779154250, 26474089989299

Offset: 1

N. J. A. Sloane

The inverse binomial transform is 1,1,3,6,..., i.e., the unsigned version of A077926. - R. J. Mathar, May 15 2008

a(n+1)/a(n) tends to a limit which is equal to the largest real root of the denominator of the g.f., 3.20556943040... = A246773 . - Robert G. Wilson v, Feb 01 2015

W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics I, p. 259.

T. D. Noe, Table of n, a(n) for n = 1..200
I. G. Enting and A. J. Guttmann, On the area of square lattice polygons, J. Statist. Phys., 58 (1990), 475-484.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 367
Dean Hickerson, Counting Horizontally Convex Polyominoes, J. Integer Sequences, Vol. 2 (1999), #99.1.8.
David A. Klarner, Some results concerning polyominoes, Fibonacci Quarterly 3 (1965), 9-20.
David A. Klarner, The number of graded partially ordered sets, Journal of Combinatorial Theory, vol.6, no.1, pp.12-19, (January-1969).
Todd Mullen, On Variants of Diffusion, Dalhousie University (Halifax, NS Canada, 2020).
R. Pemantle and M. C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, SIAM Rev., 50 (2008), no. 2, 199-272. See p. 239
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
G. Pólya, On the number of certain lattice polygons, J. Combinatorial Theory 6 1969 102--105. MR0236031 (38 #4329) - From _N. J. A. Sloane_, Jun 05 2012
K. A. Van'kov, V. M. Zhuravlyov, Regular tilings and generating functions, Mat. Pros. Ser. 3, issue 22, 2018 (127-157) [in Russian]. See page 128. - _N. J. A. Sloane_, Jan 09 2019
Kirill Vankov, Valerii Zhuravlev, Regular and semiregular (uniform) tilings and generating functions, hal-02535947, [math.CO], 2020.
Eric Weisstein's World of Mathematics, Column-Convex Polyomino.
D. Zeilberger, Automated counting of LEGO towers, arXiv:math/9801016 [math.CO], 1998.
V. M. Zhuravlev, Horizontally-convex polyiamonds and their generating functions, Mat. Pros. 17 (2013), 107-129 (in Russian).
Index entries for linear recurrences with constant coefficients, signature (5, -7, 4).

Cf. A049219, A049220 (partial sums), A049221, A049222, A246773, A273895.

I:=[1,2,6,19,61]; [n le 5 select I[n] else 5*Self(n-1)-7*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 15 2015

a[n_] := a[n] = If[n<5, {1, 2, 6, 19}[[n]], 5a[n-1] - 7a[n-2] + 4a[n-3]]; Table[a[n], {n, 30}]
Join[{1},LinearRecurrence[{5,-7,4},{2,6,19},40]] (* Harvey P. Dale, Sep 11 2014 *)
Rest@ CoefficientList[ Series[x (1 - x)^3/(1 - 5x + 7x^2 - 4x^3), {x, 0, 28}], x] (* Robert G. Wilson v, Feb 01 2015 *)

makelist(sum(sum(binomial(k,i)*binomial(n+2*i-1,4*k-i),i,0,k),k,0,n-1),n,0,24); /* Emanuele Munarini, May 19 2011 */

{a(n) = if( n<0, 0, polcoeff( x * (1 - x)^3 / (1 - 5*x + 7*x^2 - 4*x^3) + x * O(x^n), n))}; /* Michael Somos, Jun 02 2016 */

G.f.: x*(1-x)^3/(1 - 5*x + 7*x^2 - 4*x^3). - Simon Plouffe in his 1992 dissertation
a(n) = 5*a(n-1) - 7*a(n-2) + 4*a(n-3) for n >= 5.
a(n) = sum(k=0..n-1, sum(i=0..k, binomial(k,i)*binomial(n+2*i-1,4*k-i))). - Emanuele Munarini, May 19 2011
a(n) = a(n-1) + A049219(n) + A049220(n) for n >= 2.
Row sums of A273895. - Michael Somos, Jun 02 2016

More terms from Dean Hickerson

A308359 Triangle T(n,w) read by rows: the number of fixed polyominoes with n cells and width w of the convex hull.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 8, 1, 1, 18, 31, 12, 1, 1, 35, 95, 68, 16, 1, 1, 66, 269, 282, 121, 20, 1, 1, 123, 721, 1027, 638, 190, 24, 1, 1, 228, 1866, 3468, 2817, 1226, 275, 28, 1, 1, 421, 4728, 11132, 11254, 6391, 2110, 376, 32, 1, 1, 776, 11804, 34558, 42099, 29388, 12758, 3354, 493, 36, 1

Offset: 1

R. J. Mathar, May 22 2019

The sequence counts the fixed n-ominoes with prescribed bounding box width w and variable height w <= h <= n.

			T(3,2) = 4 counts the 4 variants of the L-shaped tromino rotated by multiples of 90 degrees. T(4,2) = 9 counts one O-tetromino in a 2 X 2 box, 4 L-tetrominoes in a 3 X 2 box, 2 T-tetromoes in a 3 X 2 box, and 2 Z-tetrominoes in a 3 X 2 box.
The triangle starts
  1;
  1,   1;
  1,   4,   1;
  1,   9,   8,   1;
  1,  18,  31,  12,   1;
  1,  35,  95,  68,  16,   1;
  1,  66, 269, 282, 121,  20,   1;

R. J. Mathar, Table of n, a(n) for n = 1..109
Index entries for sequences related to polyominoes

Cf. A027053 (column w=2), A335606 (w=3), A001168 (row sums), A273895, A292357 (prescribed w and h).

T(n,1) = T(n,n) = 1 (the straight n-ominoes).
T(n,n-1) = 4*n-8 for n >= 3 (width n-1 and height 2).
Conjecture: T(n,n-2) = 8*n^2 - 51*n + 86 for n >= 5.

cp's OEIS Frontend

A001169 Number of board-pile polyominoes with n cells.

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