A001169 Number of board-pile polyominoes with n cells. (Formerly M1636 N0639)

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

1,2

Comments

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

References

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.

Links

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).

Formula

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

Mathematica

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 *)

Prog

(PARI) {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 */
(Maxima) 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 */
(Magma) 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

Crossrefs

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

Sequence in context: A204200 A371708 A318127 * A187276 A022041 A018906

Adjacent sequences: A001166 A001167 A001168 * A001170 A001171 A001172

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Dean Hickerson

Status

approved