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.

A338955 Number of achiral colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 24124751133507584, 883287060208158070437496209, 27692675763559261523047959805034496, 18070082615414169898334284655914306640625, 1018202231744161700740376040914469837333037056
Offset: 1

Views

Author

Robert A. Russell, Nov 17 2020

Keywords

Comments

An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. There are 576 elements in the automorphism group of the 24-cell that are not in its rotation group. They divide into 10 conjugacy classes. The first formula is obtained by averaging the edge (or face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Count Odd Cycle Indices Count Odd Cycle Indices
12 x_1^24x_2^36 96 x_1^2x_2^2x_3^2x_6^14
12 x_1^8x_2^44 96 x_3^8x_6^12
12+12 x_3^48 96 x_2^3x_6^15
72+72 x_4^24 96 x_6^16

Links

Crossrefs

Cf. A338952 (oriented), A338953 (unoriented), A338954 (chiral), A338959 (exactly n colors), A338951 (vertices, facets), A331353 (5-cell), A331361 (8-cell edges, 16-cell faces), A331357 (16-cell edges, 8-cell faces), A338967 (120-cell, 600-cell).

Programs

  • Mathematica
    Table[(8n^16+8n^18+16n^20+12n^24+2n^48+n^52+n^60)/48,{n,15}]

Formula

a(n) = (8*n^16 + 8*n^18 + 16*n^20 + 12*n^24 + 2*n^48 + n^52 + n^60) / 48.
a(n) = Sum_{j=1..Min(n,60)} A338959(n) * binomial(n,j).
a(n) = 2*A338953(n) - A338952(n) = A338952(n) - 2*A338954(n) = A338953(n) - A338954(n).

A338956 Number of oriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

Original entry on oeis.org

1, 137548893254081168086800766, 11046328890861010626464488614428032600986342, 10897746068335468788318134977474134922662053604436974448, 21912802868317153141871319582922663027477920477404414535105616050
Offset: 1

Views

Author

Robert A. Russell, Nov 17 2020

Keywords

Comments

Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. For n>96, a(n) = 0.

Links

Crossrefs

Cf. A338957 (unoriented), A338958 (chiral), A338959 (achiral), A338952 (up to n colors), A338948 (vertices, facets), A331350 (5-cell), A331358 (8-cell edges, 16-cell faces), A331354 (16-cell edges, 8-cell faces), A338980 (120-cell, 600-cell).

Programs

  • Mathematica
    bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (* binomial series *)
Drop[CoefficientList[bp[8]/6+bp[12]/4+bp[16]/12+bp[18]/18+7bp[24]/48+bp[32]/12+bp[36]/18+19bp[48]/576+bp[50]/8+bp[96]/576,x],1]

Formula

A338952(n) = Sum_{j=1..Min(n,96)} a(n) * binomial(n,j).
a(n) = A338957(n) + A338958(n) = 2*A338957(n) - A338959(n) = 2*A338958(n) + A338959(n).

A338957 Number of unoriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

Original entry on oeis.org

1, 68774446639102959610154174, 5523164445430505754875774375105924818979901, 5448873034167734394172913824852272971748608894646534804, 10956401434158576570935668826433407535831446552957081921713485225
Offset: 1

Views

Author

Robert A. Russell, Nov 17 2020

Keywords

Comments

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. For n>96, a(n) = 0.

Links

Crossrefs

Cf. A338956 (oriented), A338958 (chiral), A338959 (achiral), A338953 (up to n colors), A338949 (vertices, facets), A063843 (5-cell), A331359 (8-cell edges, 16-cell faces), A331355 (16-cell edges, 8-cell faces), A338981 (120-cell, 600-cell).

Programs

  • Mathematica
    bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (* binomial series *)
Drop[CoefficientList[bp[8]/12+bp[12]/8+bp[16]/8+bp[18]/9+bp[20]/6+19bp[24]/96+bp[32]/24+bp[36]/36+43bp[48]/1152+bp[50]/16+bp[52]/96+bp[60]/96+bp[96]/1152,x],1]

Formula

A338953(n) = Sum_{j=1..Min(n,96)} a(n) * binomial(n,j).
a(n) = A338956(n) - A338958(n) = (A338956(n) + A338959(n)) / 2 = A338958(n) + A338959(n).

A338958 Number of chiral pairs of colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

Original entry on oeis.org

68774446614978208476646592, 5523164445430504871588714239322107782006441, 5448873034167734394145221152621861950913444709790439644, 10956401434158576570935650756489255491646473924447332613392130825
Offset: 2

Views

Author

Robert A. Russell, Nov 17 2020

Keywords

Comments

Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. For n>96, a(n) = 0.

Links

Crossrefs

Cf. A338956 (oriented), A338957 (unoriented), A338959 (achiral), A338954 (up to n colors), A338950 (vertices, facets), A331352 (5-cell), A331360 (8-cell edges, 16-cell faces), A331356 (16-cell edges, 8-cell faces), A338982 (120-cell, 600-cell).

Programs

  • Mathematica
    bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (*binomial series*)
Drop[CoefficientList[bp[8]/12+bp[12]/8-bp[16]/24-bp[18]/18-bp[20]/6-5bp[24]/96+bp[32]/24+bp[36]/36-5bp[48]/1152+bp[50]/16-bp[52]/96-bp[60]/96+bp[96]/1152,x],2]

Formula

A338954(n) = Sum_{j=2..Min(n,96)} a(n) * binomial(n,j).
a(n) = A338956(n) - A338957(n) = (A338956(n) - A338959(n)) / 2 = A338957(n) - A338959(n).
Showing 1-4 of 4 results.

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