cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A049219 Number of horizontally convex n-ominoes in which the top row has exactly 1 square.

Original entry on oeis.org

1, 1, 3, 10, 33, 107, 344, 1103, 3535, 11330, 36317, 116415, 373176, 1196243, 3834643, 12292218, 39403561, 126310851, 404898200, 1297929287, 4160602439, 13337099986, 42753000005, 137047709879, 439315949304, 1408257777387
Offset: 1

Views

Author

Dean Hickerson, Aug 10 1999

Keywords

Links

Crossrefs

Cf. A001169, A049221.

Programs

  • Mathematica
    a[ n_ ] := a[ n ]=If[ n<5, {1, 1, 3, 10}[ [ n ] ], 5a[ n-1 ]-7a[ n-2 ]+4a[ n-3 ] ]
LinearRecurrence[{5,-7,4},{1,1,3,10,33,107},40] (* Harvey P. Dale, Nov 19 2019 *)

Formula

G.f.: x (1-x)^2 (1-2x)/(1-5x+7x^2-4x^3).
a(n) = 5a(n-1) - 7a(n-2) + 4a(n-3) for n >= 5.
a(n) = A001169(n-1) + A049221(n) for n >= 2.

A049220 Number of horizontally convex n-ominoes in which the top row has at least 2 squares and the rightmost square in the top row is above the leftmost square in the second row.

Original entry on oeis.org

0, 0, 1, 3, 9, 28, 89, 285, 914, 2931, 9397, 30124, 96565, 309545, 992266, 3180775, 10196193, 32684604, 104772769, 335856389, 1076610978, 3451151243, 11062904925, 35462909836, 113678819677, 364405349233, 1168126647770
Offset: 1

Views

Author

Dean Hickerson, Aug 10 1999

Keywords

Links

Crossrefs

Cf. A001169.

Programs

  • Mathematica
    a[ n_ ] := a[ n ]=If[ n<6, {0, 0, 1, 3, 9}[ [ n ] ], 5a[ n-1 ]-7a[ n-2 ]+4a[ n-3 ] ]

Formula

G.f.: x^3 (1-x)^2/(1-5x+7x^2-4x^3).
a(n) = 5a(n-1) - 7a(n-2) + 4a(n-3) for n >= 6.
a(n) = a(n-1) + A001169(n-2) for n >= 3.

A187077 Number of row-convex polyplets with n cells.

Original entry on oeis.org

1, 4, 18, 83, 385, 1788, 8305, 38575, 179170, 832189, 3865253, 17952864, 83385309, 387298083, 1798875698, 8355202169, 38807241321, 180247221864, 837190686169, 3888482927823, 18060759310562, 83886449530197, 389625723579965
Offset: 1

Views

Author

David Bevan, Mar 03 2011

Keywords

Comments

Equivalent to a sequence of row-convex polyhexes (A059716).

Examples

			a(3) = 18 = A006770(3)-2 omits the two 3-plets with non-convex rows (V and inverted V).

Crossrefs

Cf. A006770 (all fixed polyplets); A059716 (row-convex polyhexes); A001169 (row-convex polyominoes).

Programs

  • Mathematica
    a[n_]:={1,4,18,83}[[n]]/;n<5; a[n_]:=a[n]=7a[n-1]-13a[n-2]+10a[n-3]-2a[n-4]; Array[a, 23]

Formula

G.f.: -((x(x-1)^3)/(1-7x+13x^2-10x^3+2x^4)).
a(n) = 7a(n-1)-13a(n-2)+10a(n-3)-2a(n-4) for n > 4.

A187276 Number of d+/d- diagonally convex polyominoes with n cells.

Original entry on oeis.org

1, 2, 6, 19, 61, 196, 630, 2024, 6499, 20860, 66941, 214797, 689201, 2211347, 7095226, 22765414, 73044113, 234366327, 751978494, 2412768983, 7741517800, 24839137696, 79697907919, 255715662623
Offset: 1

Views

Author

David Bevan, Mar 07 2011

Keywords

Comments

A polyomino is d+ [d-] convex if the intersection of its interior with any line of slope 1 [-1] through the centers of the cells is connected.

Examples

			A(5) = 61 = A001168(5) - 2, omitting two of the orientations of the V pentomino.

References

  • M. Bousquet-Mélou and R. Brak, "Exactly Solved Models", in A. J. Guttmann, ed., Polygons, Polyominoes and Polycubes, Springer, 2009, pp. 46 & 76.

Crossrefs

Cf. A001168 (fixed polyominoes), A001169 (row-convex polyominoes).

Programs

  • Mathematica
    ab[n_,m_,q_]:=Sum[q[n-m-r,k],{r,1,m},{k,m+1-r,n-m-r}]
bb[n_,m_,q_]:=Sum[q[n-m-r,m-r],{r,1,m-1}]+Sum[q[n-m-r,k],{r,1,m-1},{k,m-r,n-m-r}]
cb[n_,m_,q_]:=Sum[q[n-m-r,m-1-r],{r,1,m-2}]
a[n_,m_]:=0/;n<=1||m<=0
a[n_,m_]:=a[n,m]=Sum[(k-m)p[n-m,k],{k,m+1,n-m}]+ab[n,m,b]+2ab[n,m,c]+Sum[(r-1)c[n-m-r,m+1-r],{r,2,m}]
b[1,1]=1;
b[n_,m_]:=0/;n<=1||m<=0
b[n_,m_]:=b[n,m]=2Sum[p[n-m,k],{k,m,n-m}]+bb[n,m,b]+2bb[n,m,c]+2Sum[(r-1)c[n-m-r,m-r],{r,2,m-1}]
c[n_,m_]:=0/;n<=1||m<=0
c[n_,m_]:=c[n,m]=p[n-m,m-1]+cb[n,m,b]+2cb[n,m,c]+Sum[(r-1)c[n-m-r,m-1-r],{r,2,m-2}]
p[n_,m_]:=a[n,m]+b[n,m]+c[n,m]
Table[Sum[p[n,m],{m,(n+1)/2}],{n,20}]

Extensions

Typo in example corrected by David Bevan, Mar 23 2013

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

Views

Author

Jean-François Alcover, Sep 03 2014

Keywords

Comments

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

Examples

			3.20556943040059031170202861778382342637710891959769944...

Links

Crossrefs

Cf. A001169, A006958, A067675, A246772.

Programs

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

Formula

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.

A273895 T(n, k) is the number of Horizontal Convex Polyominoes with n cells and k rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 9, 8, 1, 0, 1, 16, 31, 12, 1, 0, 1, 25, 85, 68, 16, 1, 0, 1, 36, 190, 260, 121, 20, 1, 0, 1, 49, 371, 777, 604, 190, 24, 1, 0, 1, 64, 658, 1960, 2299, 1180, 275, 28, 1, 0, 1, 81, 1086, 4368, 7221, 5509, 2052, 376, 32, 1, 0
Offset: 0

Views

Author

Michael Somos, Jun 02 2016

Keywords

Examples

			Triangle begins:
0,
0, 1,
0, 1, 1,
0, 1, 4, 1,
0, 1, 9, 8, 1,

Links

Crossrefs

Cf. A001169.

Programs

  • Mathematica
    T[n_, m_] := Sum[Sum[Sum[Binomial[i - 2*j, j]*2^(i - 3*j)*Binomial[k + j, i - 2*j]*Binomial[k + 3*j - i, m + j - i - 1], {j, 0, m + i - 1}]*Binomial[ n - k - 2, n - k - i - 1], {i, 0, n - k - 1}], {k, 0, n - 1}]; Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2019, after Vladimir Kruchinin *)
  • Maxima
    T(n,m):=sum(sum((sum(binomial(i-2*j,j)*2^(i-3*j)*binomial(k+j,i-2*j)*binomial(k+3*j-i,m+j-i-1),j,0,m+i-1))*binomial(n-k-2,n-k-i-1),i,0,n-k-1),k,0,n-1); /* Vladimir Kruchinin, Jan 27 2019 */
  • PARI
    {T(n, k) = if( k<0 || k>n, 0, polcoeff( polcoeff( x * y *(1 - x)^3 / ((1 - x)^4 - x * y * (1 - x - x^2 + x^3 + x^2 * y)) + x * O(x^n), n), k))};

Formula

G.f.: x * y * (1 - x)^3 / ((1 - x)^4 - x * y * (1 - x - x^2 + x^3 + x^2 * y)) = Sum_{0<=k<=n} T(n, k) * x^n * y^k.
Row sums are A001169.
T(n,m) = Sum_{k=0..n-1} Sum_{i=0..n-k-1} [Sum_{j=0..m+i-1} C(i-2*j,j)*2^(i-3*j)*C(k+j,i-2*j)*C(k+3*j-i,m+j-i-1)]*C(n-k-2,n-k-i-1). - Vladimir Kruchinin, Jan 27 2019

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

Views

Author

Jean-François Alcover, Sep 03 2014

Keywords

Examples

			0.180915501881560609515895773010001800494429103399881...

Links

Crossrefs

Cf. A001169, A006958, A067675, A246773.

Programs

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

Formula

u = first root of 944*x^3 - 295*x^2 + 28*x - 1.
A001169(n) ~ u*v^n, where v = A246773.

A359661 a(n) is the number of free convex polyominoes of n cells.

Original entry on oeis.org

1, 1, 2, 5, 11, 29, 72, 191, 478, 1211, 2973, 7274, 17455, 41645, 98271, 230848, 539000, 1254936
Offset: 1

Views

Author

John Mason, Jan 10 2023

Keywords

Comments

A convex polyomino is such that any vertical or horizontal line connecting two points within the polyomino remains completely within the polyomino.
It has a perimeter of length equal to that of its enclosing rectangle.
A polyomino is convex if and only if (i) it is a board-pile polyomino and (ii) rotated 90 degrees it is still a board-pile-polyomino.

Crossrefs

Cf. A000105, A001169, A067675.

A061706 G.f.: x^2*(x-1)^8/(4*x^9+x^8-79*x^7+166*x^6-225*x^5+206*x^4-129*x^3+50*x^2-11*x+1).

Original entry on oeis.org

0, 0, 1, 3, 11, 44, 185, 805, 3552, 15671, 68827, 301108, 1314553, 5735237, 25024104, 109216135, 476775195, 2081568940, 9088283129, 39679791093, 173240442512, 756351086215, 3302134987931, 14416689863220, 62941426366169
Offset: 0

Views

Author

Vladeta Jovovic, Jun 18 2001

Keywords

References

  • R. P. Stanley, Enumerative Combinatorics I, 4.7.19.

Crossrefs

Cf. A001169.

Formula

a(n) = 11*a(n - 1) - 50*a(n - 2) + 129*a(n - 3) - 206*a(n - 4) + 225*a(n - 5) - 166*a(n - 6) + 79*a(n - 7) - a(n - 8) - 4*a(n - 9) for n >= 11.
Showing 1-9 of 9 results.

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