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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A047775 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type B.

Original entry on oeis.org

0, 0, 0, 0, 2, 5, 11, 25, 66, 131, 349, 708, 1911, 3856, 10604, 21597, 59961, 123266, 345060, 715198, 2015416, 4206926, 11919257, 25032840, 71246129, 150413234, 429750208, 911379241, 2612614298, 5562367173, 15991792731, 34164355260
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 B achiral symmetry and n tetrahedral cells. The plane of symmetry bisects a tetrahedral cell (321); the order of the symmetry group is 2. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 29 2024

Links

Crossrefs

Cf. A047772.
Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U), A047751 (type K), A047753 (type I), A047760 (type F), A047764 (type Q), A047765 (type P), A047773 (type D).

Programs

Formula

a(n) = (1/2)*(A047749(n) - 2*A047773(n) - 2*A047760(n) - A047753(n) - A047751(n) - A047764(n) - A047765(n)).
G.f.: (2 - G(z^4) - G(z^6))/z + (G(z^2) + z*G(z^2)^2 - G(z^4) + z*G(z^4) - z^2*G(z^4)^2 + z^2*G(z^6) + z^2*G(z^12) + z^8*G(z^12)^2) / 2 + z - z*G(z^4)^2 - z*G(z^6) - z^2*G(z^6)^2 - z^4*G(z^6)^2 + 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 29 2024

A047776 Number of chiral pairs of asymmetric dissectable polyhedra with n tetrahedral cells (type A).

Original entry on oeis.org

0, 0, 0, 0, 2, 11, 71, 370, 2005, 10682, 58167, 320116, 1789210, 10121965, 57933469, 334919626, 1953800059, 11489466014, 68053583772, 405713887061, 2433000197471, 14668527134167, 88869448492895, 540834097467624, 3304961431043989, 20273201718862728, 124798671079300720, 770762029389852807
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

One of 17 different symmetry types comprising A007173 and A027610 and one of 7 for A371350. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both asymmetric (type A) with n tetrahedral cells. The order of the symmetry group is 1. Each member of a chiral pair is a reflection but not a rotation of the other. - Robert A. Russell, Mar 31 2024

Links

Crossrefs

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047775 (type B), A047774 (type C). A047773 (type D), A047762 (type E), A047760 (type F), A047758 (type G), A047754 (type H), A047753 (type I), A047752 (type J), A047751 (type K), A047771 (type L), A047769 (type M), A047766 (type N|O), A047765 (type P), A047764 (type Q).

Programs

  • Mathematica
    Table[If[n < 5, 0, Binomial[3 n, 2 n + 2]/(3 n (n - 1))
    - If[OddQ[n], Binomial[3 n/2 - 1/2, n + 1] 3/(n - 1),
     7 Binomial[3 n/2, n + 1]/(3 n)]
    - Switch[Mod[n, 3], 1, Binomial[n - 1, 2 n/3 + 1/3]/(n - 1), 2,
     Binomial[n - 1, 2 n/3 + 2/3]/(n - 2), _, 0]
    + Switch[Mod[n, 4], 1, Binomial[3 n/4 - 3/4, n/2 + 1/2] 2/(3 (n - 1))
      + Binomial[3 n/4 + 1/4, n/2 + 3/2] 4/(n - 1) +
      Binomial[3 n/4 - 3/4, n/2 + 1/2] 4/(n + 3), 2,
     Binomial[3 n/4 - 1/2, n/2 + 1] 8/(n - 2), 3,
     Binomial[3 n/4 - 1/4, n/2 + 3/2] 12/(n - 3), 0,
     Binomial[3 n/4 - 1, n/2 + 1] 12/(n - 4)] +
    Switch[Mod[n, 6], 1, Binomial[n/2 - 1/2, n/3 + 2/3] 6/(n - 1), 2,
     Binomial[n/2 - 1, n/3 + 1/3] 4/(n - 2) +
      Binomial[n/2, n/3 + 4/3] 6/(n - 2) +
      Binomial[n/2 - 1, n/3 + 1/3] 6/(n + 4), 4,
     Binomial[n/2 - 1, n/3 + 2/3] 12/(n - 4), 5,
     Binomial[n/2 - 1/2, n/3 + 1/3] 9/(n + 4), _, 0] +
    Switch[Mod[n, 12], 2, -Binomial[n/4 - 1/2, n/6 + 2/3] 12/(n - 2), 5,
     Binomial[n/4 - 5/4, n/6 - 5/6] 2/(n + 1),
     8, -Binomial[n/4 - 1, n/6 - 1/3] 12/(n + 4), _, 0] -
    Switch[Mod[n, 24], 5, Binomial[n/8 - 5/8, n/12 - 5/12] 12/(n + 7), 17,
     Binomial[n/8 - 9/8, n/12 - 5/12] 24/(n + 7), , 0]]/2, {n, 1, 60}] (* _Robert A. Russell, Apr 09 2012 *)

Formula

From Robert A. Russell, Mar 31 2024: (Start)
a(n) = A001764(n)/(12(n+1)) - A047775(n)/2 - A047774(n)/3 - A047773(n)/6 - A047762(n)/2 - A047760(n)/4 - A047758(n)/4 - A047754(n)/4 - A047753(n)/8 - A047752(n)/12 - A047751(n)/24 - A047771(n)/2 - A047769(n)/2 - A047766(n)/6 - A047766(n)/6 - A047765(n)/4 - A047764(n)/12.
G.f.: (G(z^4) + G(z^6) - 2)/(2z) - z/3 + G(z)/6 - G(z)^2/12 + z*G(z)^4/24 - 7*G(z^2)/12 - 3z*G(z^2)^2/8 - z*G(z^3)/6 - z^2*G(z^3)^2/12 + G(z^4)/2 - z*G(z^4)/6 + (z*G(z^4)^2 + z^2*G(z^4)^2 + z*G(z^6))/2 + z^2*G(z^6)/12 + (z^2*G(z^6)^2 + z^4*G(z^6)^2 - z^2*G(z^12))/2 + z^5*G(z^12)/6 - (z^8*G(z^12)^2 + z^5*G(z^24) + z^17*G(z^24)^2)/2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)

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.
Previous Showing 21-23 of 23 results.

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

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