A006958 -id:A006958 - cp's OEIS frontend

A276994 Decimal expansion of the Klarner-Rivest polyomino constant.

2, 3, 0, 9, 1, 3, 8, 5, 9, 3, 3, 3, 0, 4, 9, 4, 7, 3, 1, 0, 9, 8, 7, 2, 0, 3, 0, 5, 0, 1, 7, 2, 1, 2, 5, 3, 1, 9, 1, 1, 8, 1, 4, 4, 7, 2, 5, 8, 1, 6, 2, 8, 4, 0, 1, 6, 9, 4, 4, 0, 2, 9, 0, 0, 2, 8, 4, 4, 5, 6, 4, 4, 0, 7, 4, 8, 3, 1, 6, 8, 4, 2, 7, 1, 7, 2, 8, 1, 6, 1, 5, 7, 7, 4, 4, 1, 2, 1, 7, 4, 3, 7, 4, 6, 1

Offset: 1

Vaclav Kotesovec, Sep 27 2016

Analytic Combinatorics (Flajolet and Sedgewick, 2009, p. 662) has a wrong value of this constant (2.309138593331230...).

			2.309138593330494731098720305017212531911814472581628401694402900284456440748...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.19 (Klarner's polyomino constant), p. 380.

E. A. Bender, Convex n-ominoes, Discrete Math., 8 (1974), 219-226.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, p. 662.
D. A. Klarner and R. L. Rivest, Asymptotic bounds for the number of convex n-ominoes, Discrete Math., 8 (1974), 31-40.

Cf. A006958.

1/z/.FindRoot[Sum[(-1)^n * z^(n*(n+1)/2) / QPochhammer[z, z, n]^2, {n, 0, 1000}], {z, 2/5}, WorkingPrecision -> 120]

Equals lim n -> infinity A006958(n)^(1/n).

1/A276994 = 0.4330619231293906645846169654189837... is the smallest positive root of the equation Sum_{n>=0} ((-1)^n * z^(n*(n+1)/2) / (Product_{k=1..n} 1-z^k)^2) = 0.

A285175 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns strictly increasing.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 5, 1, 7, 11, 1, 1, 11, 1, 13, 23, 9, 1, 7, 11, 11, 11, 25, 1, 51, 1, 1, 39, 13, 45, 23, 1, 15, 59, 25, 1, 135, 1, 41, 73, 17, 1, 9, 45, 73, 83, 61, 1, 45, 107, 63, 111, 19, 1, 135, 1, 21, 259, 1, 205, 279, 1, 85, 143, 349, 1

Offset: 1

Gus Wiseman, Feb 26 2018

nonn

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(15) = 11 tableaux:
1 2 3   1 2 4   1 3 4   1 2 5   1 3 5
4 5     3 5     2 5     3 4     2 4
.
1 2 3   1 2 3   1 2 4   1 2 4   1 3 4
2 4     3 4     2 3     3 4     2 4
.
1 2 3
2 3

Cf. A000085, A001221, A005117, A006958, A015128, A056239, A138178, A153452, A238690, A296150, A296188, A297388, A299925, A299926, A299968, A300118, A300120, A300122.

a[n_]:=If[n===1,1,Sum[a[n/q*Times@@Cases[FactorInteger[q],{p_,k_}:>If[p===2,1,NextPrime[p,-1]^k]]],{q,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Array[a,100]

A067676 Number of fixed directed convex polyominoes with n cells.

Original entry on oeis.org

1, 2, 5, 13, 33, 82, 200, 481, 1144, 2699, 6329, 14775, 34381, 79819, 185001, 428290, 990716, 2290424, 5293153, 12229209, 28249088, 65246630, 150687282, 347993954, 803620981, 1855754764, 4285319033, 9895581541, 22850547145, 52765494456, 121843455307

Offset: 1

Steven Finch, Feb 04 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..500
M. Bousquet-Mélou and J.-M. Fédou, The generating function of convex polyominoes: the resolution of a q-differential system, Discrete Math. 137 (1995) 53-75.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.

Cf. A006958, A067675 (fixed convex polyominoes), A191148 (fixed line-convex polycubes in 3 dimensions).

More terms from Sean A. Irvine, Jan 02 2024

A246773 Decimal expansion of 'v', an auxiliary constant associated with the asymptotic number of row-convex polyominoes.

Original entry on oeis.org

3, 2, 0, 5, 5, 6, 9, 4, 3, 0, 4, 0, 0, 5, 9, 0, 3, 1, 1, 7, 0, 2, 0, 2, 8, 6, 1, 7, 7, 8, 3, 8, 2, 3, 4, 2, 6, 3, 7, 7, 1, 0, 8, 9, 1, 9, 5, 9, 7, 6, 9, 9, 4, 4, 0, 4, 7, 0, 5, 5, 2, 2, 0, 3, 5, 5, 1, 8, 3, 4, 7, 9, 0, 3, 5, 9, 1, 6, 7, 4, 6, 9, 1, 7, 6, 4, 1, 8, 2, 6, 9, 5, 7, 8, 0, 5, 2, 5, 0, 7, 8, 4, 9, 9

Offset: 1

Jean-François Alcover, Sep 03 2014

Essentially the same digit sequence as A137421. - R. J. Mathar, Sep 06 2014

			3.20556943040059031170202861778382342637710891959769944...

Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 48.

Cf. A001169, A006958, A067675, A246772.

v = Root[x^3 - 5*x^2 + 7*x - 4, x, 1]; RealDigits[v, 10, 104] // First

v = first root of x^3 - 5*x^2 + 7*x - 4 = (x-2)^3+(x-2)^2-(x-2)-2.

A001169(n) ~ u*v^n, where u = A246772.

A225114 Number of skew partitions of n whose diagrams have no empty rows and columns.

Original entry on oeis.org

1, 1, 3, 9, 28, 87, 272, 850, 2659, 8318, 26025, 81427, 254777, 797175, 2494307, 7804529, 24419909, 76408475, 239077739, 748060606, 2340639096, 7323726778, 22915525377, 71701378526, 224349545236, 701976998795, 2196446204672, 6872555567553, 21503836486190, 67284284442622, 210528708959146

Offset: 0

Joerg Arndt, Apr 29 2013

nonn

A skew partition S of size n is a pair of partitions [p1,p2] where p1 is a partition of the integer n1, p2 is a partition of the integer n2, p2 is an inner partition of p1, and n=n1-n2. We say that p1 and p2 are respectively the inner and outer partitions of S. A skew partition can be depicted by a diagram made of rows of cells, in the same way as a partition. Only the cells of the outer partition p1 which are not in the inner partition p2 appear in the picture. [from the Sage manual, see links]

			The a(4)=28 skew partitions of 4 are
01:  [[4], []]
02:  [[3, 1], []]
03:  [[4, 1], [1]]
04:  [[2, 2], []]
05:  [[3, 2], [1]]
06:  [[4, 2], [2]]
07:  [[2, 1, 1], []]
08:  [[3, 2, 1], [1, 1]]
09:  [[3, 1, 1], [1]]
10:  [[4, 2, 1], [2, 1]]
11:  [[3, 3], [2]]
12:  [[4, 3], [3]]
13:  [[2, 2, 1], [1]]
14:  [[3, 3, 1], [2, 1]]
15:  [[3, 2, 1], [2]]
16:  [[4, 3, 1], [3, 1]]
17:  [[2, 2, 2], [1, 1]]
18:  [[3, 3, 2], [2, 2]]
19:  [[3, 2, 2], [2, 1]]
20:  [[4, 3, 2], [3, 2]]
21:  [[1, 1, 1, 1], []]
22:  [[2, 2, 2, 1], [1, 1, 1]]
23:  [[2, 2, 1, 1], [1, 1]]
24:  [[3, 3, 2, 1], [2, 2, 1]]
25:  [[2, 1, 1, 1], [1]]
26:  [[3, 2, 2, 1], [2, 1, 1]]
27:  [[3, 2, 1, 1], [2, 1]]
28:  [[4, 3, 2, 1], [3, 2, 1]]

Sage Development Team, Skew Partitions, Sage Reference Manual.

\\ The following program is significantly faster.
A225114(n)=
{
    my( C=vector(n, j, 1) );
    my(m=n, z, t, ret);
    while ( 1,  /* for all compositions C[1..m] of n */
\\        print( vector(m, n, C[n] ) ); /* print composition */
        t = prod(j=2,m, min(C[j-1], C[j]) + 1 );  /* A225114 */
\\        t = prod(j=2,m, min(C[j-1], C[j]) + 0 );  /* A006958 */
\\        t = prod(j=2,m, C[j-1] + C[j] + 0 );  /* A059716 */
\\        t = prod(j=2,m, C[j-1] + C[j] + 1 );  /* A187077 */
\\        t = sum(j=2,m, C[j-1] > C[j] );  /* A045883 */
        ret += t;
        if ( m<=1, break() ); /* last composition? */
        /* create next composition: */
        C[m-1] += 1;
        z = C[m];
        C[m] = 1;
        m += z - 2;
    );
    return(ret);
}
for (n=0, 30, print1(A225114(n),", "));
\\ Joerg Arndt, Jul 09 2013

[SkewPartitions(n).cardinality() for n in range(16)]

Conjectured g.f.: 1/(2 - 1/(1 - x/(1 - x/(1 - x^2/(1 - x^2/(1 - x^3/(1 - x^3/(1 - ...)))))))). - Mikhail Kurkov, Sep 03 2024

Edited by Max Alekseyev, Dec 22 2015

A227045 G.f.: 1/(1 - q/G(0)) where G(k) = 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ).

Original entry on oeis.org

1, 1, 2, 5, 13, 35, 95, 260, 713, 1959, 5386, 14815, 40759, 112151, 308609, 849240, 2337009, 6431246, 17698332, 48704714, 134032593, 368850417, 1015056867, 2793383746, 7687248186, 21154913043, 58217239536, 160210872557, 440892153268, 1213312738702, 3338974845151, 9188688696438

Offset: 0

Joerg Arndt, Jul 06 2013

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A006958 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A226729 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A226728 (g.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A227309 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).
Cf. A227310 (g.f.: 1/G(0), where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ) ).

nmax = 40; CoefficientList[Series[1/(1 - x/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 2] - Floor[Range[nmax + 2]/2])]]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ) );
gf = 1 /(1- q/G(0));
Vec(gf)

G.f.: 1/(1-q/ (1-q/(1-q/ (1-q^2/(1-q^2/ (1-q^3/(1-q^3/ (1-q^4/(1-q^4/ (1-q^5/(1-q^5/ (1-...))))))))))) ).
G.f. A(x) = 1/(1 - B(x)) where B(x) is the g.f. of A006958.
a(n) ~ c * d^n, where d = 2.751949072495748078279227332764623096815571855905843246297955690122791154... and c = 0.215973947378529032758849789768859077066690378163074586384819930605436492... - Vaclav Kotesovec, Sep 05 2017

A246772 Decimal expansion of 'u', an auxiliary constant associated with the asymptotic number of row-convex polyominoes.

Original entry on oeis.org

1, 8, 0, 9, 1, 5, 5, 0, 1, 8, 8, 1, 5, 6, 0, 6, 0, 9, 5, 1, 5, 8, 9, 5, 7, 7, 3, 0, 1, 0, 0, 0, 1, 8, 0, 0, 4, 9, 4, 4, 2, 9, 1, 0, 3, 3, 9, 9, 8, 8, 1, 0, 0, 0, 4, 9, 9, 5, 9, 4, 8, 3, 2, 4, 4, 3, 8, 9, 8, 1, 7, 8, 0, 8, 2, 4, 5, 6, 3, 2, 8, 6, 5, 8, 4, 2, 9, 4, 6, 2, 4, 4, 0, 7, 4, 9, 0, 4, 9, 1, 1, 5, 5

Offset: 0

Jean-François Alcover, Sep 03 2014

			0.180915501881560609515895773010001800494429103399881...

Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 48.

Cf. A001169, A006958, A067675, A246773.

u = Root[944*x^3 - 295*x^2 + 28*x - 1, x, 1]; RealDigits[u, 10, 103] // First

u = first root of 944*x^3 - 295*x^2 + 28*x - 1.

A001169(n) ~ u*v^n, where v = A246773.

A301412 G.f. A(x) satisfies: A(x) = 1/(1 - xA(x)/(1 - xA(x)/(1 - x^2A(x)/(1 - x^2A(x)/(1 - x^3A(x)/(1 - x^3A(x)/(1 - ...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 3, 11, 46, 205, 957, 4614, 22803, 114898, 588052, 3048612, 15975922, 84489890, 450363757, 2417104782, 13050778500, 70841037919, 386357165119, 2116097719571, 11634392901981, 64188480019008, 355255552604237, 1971866447509917, 10973973061151433, 61222237473973758

Offset: 0

Ilya Gutkovskiy, Mar 20 2018

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 11*x^3 + 46*x^4 + 205*x^5 + 957*x^6 + 4614*x^7 + 22803*x^8 + 114898*x^9 + ...

Cf. A006958, A088368, A192728.

A291378 Expansion of the series reversion of -1 + 1/(1 - x/(1 - x/(1 - x^2/(1 - x^2/(1 - x^3/(1 - x^3/(1 - ...))))))), a continued fraction.

Original entry on oeis.org

1, -2, 4, -9, 24, -74, 251, -902, 3359, -12802, 49588, -194445, 770099, -3076129, 12380317, -50162386, 204475572, -838014584, 3451174777, -14274905490, 59276495017, -247019567936, 1032709501505, -4330122550717, 18204993223606, -76728300335664, 324125242867935, -1372110743864550

Offset: 1

Ilya Gutkovskiy, Aug 23 2017

sign

Reversion of g.f. for A006958.

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Series Reversion
Index entries for reversions of series

Cf. A006958.

Rest[CoefficientList[InverseSeries[Series[-1 + 1/(1 + ContinuedFractionK[-x^Floor[(i + 1)/2], 1, {i, 1, nmax}]), {x, 0, 28}], x], x]]

G.f. A(x) satisfies: -1 + 1/(1 - A(x)/(1 - A(x)/(1 - A(x)^2/(1 - A(x)^2/(1 - A(x)^3/(1 - A(x)^3/(1 - ...))))))) = x.

a(n) ~ (-1)^(n+1) * c * d^n / n^(3/2), where d = 4.473956977950366804747779231113352537187229544... and c = 0.1202474525564857621186593278823505223773725... - Vaclav Kotesovec, May 07 2024

A291875 Expansion of 1 - x/(1 - x/(1 - x^2/(1 - x^2/(1 - x^3/(1 - x^3/(1 - x^4/(1 - x^4/ ...))))))), a continued fraction.

Original entry on oeis.org

1, -1, -1, -1, -2, -3, -6, -10, -19, -34, -63, -115, -213, -391, -723, -1333, -2463, -4547, -8403, -15522, -28686, -53006, -97963, -181042, -334606, -618415, -1142994, -2112545, -3904592, -7216810, -13338856, -24654268, -45568784, -84225393, -155675230

Offset: 0

Seiichi Manyama, Sep 04 2017

sign

Cf. A006958, A227309, A291148 (similar sequence).

a(n) = -A227309(n-1) for n > 0.

cp's OEIS Frontend

A276994 Decimal expansion of the Klarner-Rivest polyomino constant.

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A285175 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns strictly increasing.

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Mathematica

A067676 Number of fixed directed convex polyominoes with n cells.

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Extensions

A246773 Decimal expansion of 'v', an auxiliary constant associated with the asymptotic number of row-convex polyominoes.

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Mathematica

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A225114 Number of skew partitions of n whose diagrams have no empty rows and columns.

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PARI

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Extensions

A227045 G.f.: 1/(1 - q/G(0)) where G(k) = 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ).

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PARI

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A246772 Decimal expansion of 'u', an auxiliary constant associated with the asymptotic number of row-convex polyominoes.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A301412 G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x)/(1 - x*A(x)/(1 - x^2*A(x)/(1 - x^2*A(x)/(1 - x^3*A(x)/(1 - x^3*A(x)/(1 - ...))))))), a continued fraction.

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A291378 Expansion of the series reversion of -1 + 1/(1 - x/(1 - x/(1 - x^2/(1 - x^2/(1 - x^3/(1 - x^3/(1 - ...))))))), a continued fraction.

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A301412 G.f. A(x) satisfies: A(x) = 1/(1 - xA(x)/(1 - xA(x)/(1 - x^2A(x)/(1 - x^2A(x)/(1 - x^3A(x)/(1 - x^3A(x)/(1 - ...))))))), a continued fraction.