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 1-4 of 4 results.

A047756 Duplicate of A047767.

Original entry on oeis.org

0, 0, 1, 2, 2, 7, 12, 29, 55, 143, 271, 728, 1428, 3873, 7752, 21318, 43256, 120175
Offset: 0

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Author

Keywords

A047753 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type I.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 7, 0, 0, 0, 12, 0, 0, 0, 29, 0, 0, 0, 55, 0, 0, 0, 143, 0, 0, 0, 271, 0, 0, 0, 728, 0, 0, 0, 1428, 0, 0, 0, 3873, 0, 0, 0, 7752, 0, 0, 0, 21318, 0, 0, 0, 43256, 0, 0, 0, 120175, 0, 0, 0, 246675, 0, 0, 0
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

One of 17 different symmetry types comprising A007173 and A027610 and one of 10 for A371351. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type I achiral symmetry and n tetrahedral cells. The center of symmetry is the center of a tetrahedral cell (3.1); the order of the symmetry group is 8. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

Links

Crossrefs

Cf. A047767.
Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047751 (type K), A047749 (type U).

Programs

  • Mathematica
    Table[Switch[Mod[n,8],1,4Binomial[(3n-3)/8,(n-1)/8]/(n+3)-If[17==Mod[n,24],24Binomial[(n-9)/8,(n-17)/24]/(n+7),0],5,4Binomial[(3n-7)/8,(n+3)/8]/(n-1)-If[5==Mod[n,24],12Binomial[(n-5)/8,(n-5)/12]/(n+7),0],,0]-Boole[1==n],{n,50}] (* _Robert A. Russell, Mar 22 2024 *)

Formula

If n=4m+1 then A047749(m) - A047751(n), otherwise 0.
G.f.: z*G(z^8) + z^5*G(z^8)^2 - z - z^5*G(z^24) - z^17*G(z^24)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024

A047765 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type P.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 0, 7, 0, 12, 0, 29, 0, 55, 0, 143, 0, 271, 0, 728, 0, 1428, 0, 3873, 0, 7752, 0, 21318, 0, 43256, 0, 120175, 0, 246675, 0, 690678, 0, 1430715, 0, 4032015, 0, 8414610, 0, 23841480, 0, 50067108, 0, 142498637, 0, 300830572
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

One of 17 different symmetry types comprising A007173 and A027610 and one of 10 for A371351. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type P achiral symmetry and n tetrahedral cells. The center of symmetry is the altitude of a tetrahedral face (21); the order of the symmetry group is 4. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

Links

Crossrefs

Cf. A047767.
Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U), A047764 (type Q).

Programs

  • Mathematica
    Table[If[OddQ[n],0,If[OddQ[n/2],2Binomial[(3n-2)/4,(n-2)/4],Binomial[3n/4,n/4]]/(n/2+1)-Switch[Mod[n,12],2,6Binomial[(n-2)/4,(n-2)/12],8,12Binomial[(n-4)/4,(n-2)/6],,0]/(n+4)],{n,52}] (* _Robert A. Russell, Mar 22 2024 *)

Formula

If n=2m then A047749(m) - A047764(n), otherwise 0.
G.f.: G(z^4) + z^2*G(z^4)^2 - z^2*G(z^12) - z^8*G(z^12)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024

A121180 Alkane systems (see Cyvin reference for precise definition).

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 6, 26, 32, 133, 176, 708, 952, 3861, 5302, 21604, 29960
Offset: 1

Views

Author

N. J. A. Sloane, Aug 17 2006

Keywords

Comments

Appears to be A047774 without every third term (all omitted terms are zeros). - Andrey Zabolotskiy, Jul 29 2023

Links

Crossrefs

Cf. other columns of Cyvin et al.'s Table 5: A027610 (spectral isomers), A007173 (stereoisomers), A047775 (C_s), A047772 (C_i), A047774 (C_3, apparently), A047767 (C_{2h}), A047761 (C_{2v}), A047773 (C_{3v}, apparently).
Cf. A121181, A121189.
Showing 1-4 of 4 results.

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

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