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A006958
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Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused).
(Formerly M1175)
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33
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1, 2, 4, 9, 20, 46, 105, 242, 557, 1285, 2964, 6842, 15793, 36463, 84187, 194388, 448847, 1036426, 2393208, 5526198, 12760671, 29466050, 68041019, 157115917, 362802072, 837759792, 1934502740, 4467033943, 10314998977, 23818760154, 55000815222, 127004500762
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OFFSET
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1,2
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COMMENTS
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Same as: number of skew Ferrers diagrams. - Joerg Arndt, Mar 18 2014
A coin fountain is an arrangement of coins in numbered rows such that the bottom row (row 0) contains contiguous coins and such that each coin in a higher row touches exactly two coins in the next lower row. See A005169. a(n) equals the number of coin fountains with exactly n coins in the even-numbered rows of the fountain. See the illustration in the Links section. See A161492 for a refinement of this sequence. - Peter Bala, Jul 20 2019
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 1..2752 (terms 1..1000 from Robert Israel)
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.
Peter Bala, Illustration for the initial terms of the sequence
Peter Bala, Fountains of coins and skew Ferrers diagrams
E. A. Bender, Convex n-ominoes, Discrete Math., 8 (1974), 219-226.
M. P. Delest, J. M. Fedou, Enumeration of skew Ferrers diagrams, Discrete Mathematics, vol. 112 (1993), no. 1-3, pp. 65-79.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
P. Flajolet, Email to N. J. A. Sloane & S. Plouffe, Aug. 1991
P. Flajolet, Polya Festoons, INRIA Research Report, No 1507, September 1991. 6pp.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 661
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.
D. A. Klarner and R. L. Rivest, Asymptotic bounds for the number of convex n-ominoes, Discrete Math., 8 (1974), 31-40.
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FORMULA
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G.f.: 1+A(x) = 1/(1-x/(1-x/(1-x^2/(1-x^2/(1-x^3/(1-x^3/(1-...)))))) (continued fraction). - Paul D. Hanna, May 14 2005
The continued fraction given by P. Flajolet, "Polya Festoons", is derived from a q-expansion, C(x, y;q), where C(1, 1;q) = q/(1-2*q-q^3/(1-2*q^2-q^5/(1-2*q^3-q^7/(1-2*q^4-q^9/(1-...))))) = q + 2*q^2 + 4*q^3 + 9*q^4 + 20*q^5 + 46*q^6 + 105*q^7 + ... - Paul D. Hanna, May 14 2005
G.f.: 1/x/G(0) -1/x, where G(k)= 1 - x^(k+1)/(1 - x^(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013
G.f.: W(0)/x - 1/x, where W(k) = 1 - x^(k+1)/( x^(k+1) - 1/(1 - x^(k+1)/( x^(k+1) - 1/W(k+1) ))); R=1 (continued fraction). - Sergei N. Gladkovskii, Aug 27 2013
a(n) ~ c * d^n, where d = A276994 = 2.309138593330494731098720305017212531911814472581628401694402900284456440748..., c = 0.29745350581112195107675842441785013227507248969090226252518932405713367... . - Vaclav Kotesovec, Sep 21 2016
From Peter Bala, Jul 21 2019: (Start)
O.g.f. as a ratio of q-series: 1 + A(q) = N(q)/D(q) = 1 + q + 2*q^2 + 4*q^3 + ..., where N(q) = Sum_{n >= 0} (-1)^n*q^((n^2 + 3*n)/2)/Product_{k = 1..n} (1 - q^k)^2 and D(q) = Sum_{n >= 0} (-1)^n*q^((n^2 + n)/2)/Product_{k = 1..n} (1 - q^k)^2.
The constant d = 2.30913... in the above asymptotic formula is a zero of D(q) (as is 1/d).
Continued fraction representations for the o.g.f.:
1 + A(q) = 1/(1 - q/(1 - q/(1 + q*(1 - q) - q/( 1 + q*(1 - q^2) - q/(1 + q*(1 - q^3) - (...) ))))).
1 + A(q) = 1/(1 - q - q^2/(1 - q*(1 + q) - q^4/(1 - q^2*(1 + q) - q^6/(1 - q^3(1 + q) - q^8/( (...) ))))).
1 + A(q) = 1/(1 - q - q^2/(1 - q^2 - q/(1 - q^3 - q^5/(1 - q^4 - q^2/(1 - q^5 - q^8/ (1 - q^6 - q^3/(1 - q^7 - q^11/(1 - q^8 - (...) )))))))). (End)
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EXAMPLE
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G.f. may be expressed by the continued fraction: 1/(1-x/(1-x/(1-x^2/(1-x^2/(1-x^3/(1-x^3/(1-x^4/(1-...))))))))) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 105*x^7 + ...
From Michael B. Porter, Sep 21 2016; corrected by Riccardo Moschetti, Aug 11 2017: (Start)
Here are the nine parallelogram polyominoes with 4 cells, i.e., polygons convex according to the -45-degree direction, according to "Polya Festoons" of P. Flajolet:
_ _ _
_ _ _ /_ / /_ /_ /
_ /_ /_ / /_ /_ / /_ / _ _ _ _
/_ /_ / /_ / /_ / /_ /_ /_ /_ /
_
_ /_ / _ _ _ _ _
/_ / /_ / /_ /_ /_ / _ /_ /_ /
_ /_ / /_ / /_ / _ _ /_ / /_ /_ /
/_ /_ / /_ / /_ /_ /_ /
(End)
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MAPLE
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N:= 100: # to get a(1) to a(N)
M:= ceil(sqrt(N+1)):
C:= 1:
for j from M to 1 by -1 do C:= 1/(1-x^j/(1-x^j*C)) od:
S:= series(C, x, N+1):
seq(coeff(S, x, j), j=1..N); # Robert Israel, Sep 20 2016
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MATHEMATICA
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NN = 100; (* to get a(1) to a(NN) *) M = Ceiling[Sqrt[NN+1]]; c = 1; For[j = M, j >= 1, j--, c = 1/(1-x^j/(1-x^j*c))]; c = Series[c, {x, 0, NN+1}]; CoefficientList[c, x][[2 ;; NN+1]] (* Jean-François Alcover, Sep 27 2016, adapted from Robert Israel's Maple code *)
nmax = 40; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 1] - Floor[Range[nmax + 1]/2])]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)
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PROG
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(PARI) {a(n)=local(CF=1+x*O(x^n), m); for(k=0, n\2, m=n\2-k+1; CF=(1-x^((m+1)\2)/CF)); polcoeff(1/CF, n)} \\ Paul D. Hanna, May 14 2005
(PARI) /* From the Delest/Fedou reference: */
N=44; q='q+O('q^N); t=1;
qn(n) = prod(k=1, n, 1-q^k );
nm = sum(n=0, N, (-1)^n* q^(n*(n+1)/2) / ( qn(n) * qn(n+1) ) * (t*q)^(n+1) );
dn = sum(n=0, N, (-1)^n* q^(n*(n-1)/2) / ( qn(n)^2 ) * (t*q)^n );
Vec(nm/dn) \\ Joerg Arndt, Mar 18 2014
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CROSSREFS
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Cf. A075125, A276994, A275761, A275762, A161492.
Sequence in context: A000632 A090245 A274965 * A036617 A007902 A057417
Adjacent sequences: A006955 A006956 A006957 * A006959 A006960 A006961
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane, Simon Plouffe
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EXTENSIONS
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More terms from Paul D. Hanna, May 14 2005
Definition modified by Don Knuth, Sep 20 2016
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STATUS
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approved
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