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.

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A038174 Number of "polyspheres", or "connected animals" formed from n rhombic dodecahedra (or edge-connected cubes) in the f.c.c. lattice, allowing translation and rotations of the lattice, reflections and 180 deg. rotations about a 3-fold symmetry axis of the lattice.

Original entry on oeis.org

1, 1, 4, 25, 210, 2209, 24651, 284768, 3360995, 40328652, 490455189
Offset: 1

Views

Author

Achim Flammenkamp, Torsten Sillke (TORSTEN.SILLKE(AT)LHSYSTEMS.COM)

Keywords

Links

Crossrefs

Cf. A000162, A038119, A038168, A038169, A038170, A038171, A038172, A038173.

Extensions

a(9) and a(10) from Achim Flammenkamp Feb 15 1999
a(11) from Ishino Keiichiro's website added by Andrey Zabolotskiy, Mar 03 2023

A066288 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 24.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0
Offset: 1

Views

Author

Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002

Keywords

Comments

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 18 2024

References

  • W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Crossrefs

See A000162, A066273, A066281, A066283, A066287, A066453, A066454.

Formula

a(n) = 2*R(n)+G(n), see Lunnon paper for naming convention. - John Mason, Sep 18 2024

Extensions

Name clarified and more terms from John Mason, Sep 18 2024

A002881 Number of simple imperfect squared rectangles of order n up to symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 9, 34, 104, 283, 953, 3029, 9513, 30359, 98969, 323646, 1080659, 3668432, 12608491, 43745771, 153812801
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size, the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle. The order of a squared rectangle is the number of its constituent squares. [Geoffrey H. Morley, Oct 17 2012]

References

  • C. J. Bouwkamp, personal communication.
  • 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).
  • W. T. Tutte, Squaring the Square, in M. Gardner's "Mathematical Games" column in Scientific American 199, Nov. 1958, pp. 136-142, 166, Reprinted with addendum and bibliography in the US in M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles & Diversions, Simon and Schuster, New York (1961), pp. 186-209, 250 [sequence on p. 207], and in the UK in M. Gardner, More Mathematical Puzzles and Diversions, Bell (1963) and Penguin Books (1966), pp. 146-164, 186-187 [sequence on p. 162].

Links

Crossrefs

Cf. A006983, A002962, A002839, A220165.
Cf. A181735, A217153, A217154, A217156.

Formula

a(n) = A002962(n) + A220165(n).

Extensions

Edited ("simple" added to the definition, definition of "simple" given in the comments), terms a(13), a(15), a(16), a(17), and a(18) corrected, and terms extended to a(20) by Stuart E Anderson, Mar 09 2011
a(16)-a(20) corrected (excess compounds removed) by Stuart E Anderson, Apr 10 2011
Sequence reverted to the one in Bouwkamp et al. (1960), Gardner (1961), Sloane (1973), and Sloane & Plouffe (1995), which includes simple imperfect squares, by Geoffrey H. Morley, Oct 17 2012
a(19)-a(20) corrected, a(21)-a(24) added by Stuart E Anderson, Dec 03 2012
a(25) from Stuart E Anderson, May 07 2024
a(26) from Stuart E Anderson, Jul 28 2024

A038171 Number of "connected animals" formed from n 6-gon connected truncated octahedra (or corner connected cubes) in the b.c.c. lattice, allowing translation and rotations of the lattice and reflections.

Original entry on oeis.org

1, 1, 3, 12, 61, 407, 3226, 28335, 262091, 2501168, 24328920, 239931556
Offset: 1

Views

Author

Achim Flammenkamp

Keywords

Links

Crossrefs

Cf. A000162, A038119, A038168-A038174.
35th row of A366766.

Extensions

a(10) and a(11) from Joerg Arndt and Márk Péter Légrádi, Apr 30 2023

A066273 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 4, 0, 3, 24, 4, 18, 151, 25, 136, 992, 184, 938, 6769, 1300, 6792, 47469
Offset: 1

Views

Author

Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002

Keywords

Comments

This sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 04 2024

References

  • W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Crossrefs

See A000162, A066281, A066283, A066287, A066288, A066453, A066454.

Formula

a(n) = 2*D(n)+H(n)+FF(n), see Lunnon paper for naming convention. - John Mason, Sep 04 2024

Extensions

Name clarified by John Mason, Sep 04 2024
More terms from John Mason, Sep 18 2024

A066281 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 5, 11, 11, 24, 35, 84, 92, 174, 254, 606, 658, 1255, 1769, 4353, 4667, 9131
Offset: 1

Views

Author

Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002

Keywords

Comments

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 18 2024

References

  • W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Crossrefs

See A000162, A066273, A066283, A066287, A066288, A066453, A066454.

Formula

a(n) = 2*A(n)+2*BB(n)+2*BC(n)+AE(n)+BFF(n)+CJ(n)+EEE(n)+EF(n)+EFF(n), see Lunnon paper for naming convention. - John Mason, Sep 18 2024

Extensions

Name clarified and more terms from John Mason, Sep 18 2024

A066283 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 5, 2, 4, 3, 8, 4, 28, 14, 20, 20, 41
Offset: 1

Views

Author

Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002

Keywords

Comments

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 18 2024

References

  • W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Crossrefs

See A000162, A066273, A066281, A066287, A066288, A066453, A066454.

Formula

a(n) = 2*CD(n)+CF(n), see Lunnon paper for naming convention. - John Mason, Sep 18 2024

Extensions

Name clarified and more terms from John Mason, Sep 18 2024

A066287 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 8.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 1, 2, 4, 2, 5, 7, 8, 5, 10, 17, 20, 12, 23, 42, 48, 30, 59, 108
Offset: 1

Views

Author

Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002

Keywords

Comments

The entry a(14)=2 does not match the examples in A000162, which propose a(14)=5. - R. J. Mathar, Sep 03 2024
The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 04 2024

References

  • W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Crossrefs

See A000162, A066273, A066281, A066283, A066288, A066453, A066454.

Formula

a(n) = 2*AB(n)+BBC(n), see Lunnon paper for naming convention. - John Mason, Sep 04 2024

Extensions

a(14) corrected (after comment from R. J. Mathar), a(15)-a(24) added, and name clarified by John Mason, Sep 04 2024

A066453 Number of 3-dimensional polyominoes (or polycubes) with n cells and trivial rotational symmetry group.

Original entry on oeis.org

0, 0, 0, 1, 17, 127, 941, 6662, 47771, 344708, 2518713, 18585455, 138434899, 1039401564, 7859310749, 59794417068, 457408090798, 3516003907738, 27144132395911, 210375321159360, 1636229683680890, 12766881894462441
Offset: 1

Views

Author

Brendan Owen (brendan_owen(AT)yahoo.com), Dec 27 2001

Keywords

Comments

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 19 2024

References

  • W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Crossrefs

Cf. A000162, A066454, A066273, A066281, A066283, A066287, A066288.

Formula

a(n) = 2*I(n)+E(n)+F(n)+K(n), see Lunnon paper for naming convention. - John Mason, Sep 19 2024

Extensions

Name clarified by and more terms from John Mason, Sep 19 2024

A066454 Number of 3-dimensional polyominoes (or polycubes) with n cells and rotational symmetry group of order exactly 2.

Original entry on oeis.org

0, 0, 1, 4, 10, 34, 71, 246, 522, 1783, 3765, 12858, 27496, 94525, 203318, 702789, 1522315, 5290592, 11519095, 40214441, 87947507, 308236670
Offset: 1

Views

Author

Brendan Owen (brendan_owen(AT)yahoo.com), Dec 27 2001

Keywords

Comments

The sequence counts "one-sided" polycubes (A000162); chiral polycubes count twice. - John Mason, Sep 18 2024

References

  • W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Crossrefs

Cf. A000162, A066453, A066273, A066281, A066283, A066287, A066288.

Formula

a(n) = 2*B(n)+2*C(n)+J(n)+BE(n)+BF(n)+CE(n)+CK(n)+EE(n), see Lunnon paper for naming convention. - John Mason, Sep 18 2024

Extensions

Name clarified and more terms from John Mason, Sep 18 2024
Previous Showing 11-20 of 74 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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