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 1-5 of 5 results.

A338966 Number of chiral pairs of colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using subsets of a set of n colors.

Original entry on oeis.org

92307499707128546879177569498768, 124792381938502167387269721273817892704188259502965515, 122697712831832245109951209382504597654581237223625701047064169830144
Offset: 2

Views

Author

Robert A. Russell, Dec 04 2020

Keywords

Comments

Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual.
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide formulas here.
For the 600 facets of the 600-cell (vertices of the 120-cell), the formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 - 816*n^60 - 1440*n^66 + 40*n^100 - 800*n^104 - 1200*n^114 + 624*n^120 + 60*n^150 - 1800*n^152 + 40*n^200 + 400*n^208 - 59*n^300 + 450*n^302 - 60*n^330 + n^600) / 14400.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the formula is (960 n^24 + 1440 n^36 + 960 n^48 + 1200 n^60 + 336 n^72 - 1152 n^76 - 1440 n^84 - 760 n^120 - 1200 n^132 + 336 n^144 + 288 n^152 + 60 n^180 - 1800 n^182 + 440 n^240 - 59 n^360 + 450 n^364 - 60 n^396 + n^720) / 14400.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 - 816*n^120 - 1440*n^128 + 40*n^200 - 800*n^202 - 1200*n^216 + 624*n^240 + 60*n^300 - 1800*n^302 + 40*n^400 + 400*n^404 - 59*n^600 + 450*n^604 - 60*n^640 + n^1200) / 14400.

Links

Crossrefs

Cf. A338964 (oriented), A338965 (unoriented), A338967 (achiral), A338982 (exactly n colors), A000389 (5-cell), A337954 (8-cell vertices, 16-cell facets), A234249 (16-cell vertices, 8-cell facets), A338950 (24-cell).

Programs

  • Mathematica
    Table[(960n^4 +1440n^6 +960n^8 +1200n^10 +336n^12 +288n^16 -1440n^17 -1440n^19 +40n^20 +400n^22 -1200n^23 +336n^24 -1200n^27 +60n^30 -1800n^31 +288n^32 +40n^40 +400n^44 +n^60 -60n^61 +450n^62 -60n^75 +n^120)/14400, {n,2,10}]

Formula

a(n) = (960*n^4 + 1440*n^6 + 960*n^8 + 1200*n^10 + 336*n^12 + 288*n^16 - 1440*n^17 - 1440*n^19 + 40*n^20 + 400*n^22 - 1200*n^23 + 336*n^24 - 1200*n^27 + 60*n^30 - 1800*n^31 + 288*n^32 + 40*n^40 + 400*n^44 + n^60 - 60*n^61 + 450*n^62 - 60*n^75 + n^120) / 14400.
a(n) = Sum_{j=2..Min(n,120)} A338982(n) * binomial(n,j).
a(n) = A338964(n) - A338965(n) =(A338964(n) - A338967(n)) / 2 = A338965(n) - A338967(n).

A338980 Number of oriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

Original entry on oeis.org

0, 1, 184614999414571937405905419562270, 249584763877004334779054488506782340719383629107224173, 245395425663663491880846922641400894840783985813370231599231766603156
Offset: 0

Views

Author

Robert A. Russell, Dec 13 2020

Keywords

Comments

Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. For n>120, a(n) = 0.
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide generating functions here using bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k.
For the 600 facets of the 600-cell (vertices of the 120-cell), the generating function is 2*bp(20)/15 + bp(30)/5 + 2*bp(40)/15 + bp(50)/6 + 13*bp(60)/150 + bp(100)/180 + bp(104)/18 + 13*bp(120)/150 + bp(150)/120 + bp(200)/180 + bp(208)/18 + bp(300)/7200 + bp(302)/16 + bp(600)/7200.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is 2*bp(24)/15 + bp(36)/5 + 2*bp(48)/15 + bp(60)/6 + 7*bp(72)/150 + bp(76)/25 + 11*bp(120)/180 + 7*bp(144)/150 + bp(152)/25 + bp(180)/120 + 11*bp(240)/180 + bp(360)/7200 + bp(364)/16 + bp(720)/7200.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is 2*bp(40)/15 + bp(60)/5 + 2*bp(80)/15 + bp(100)/6 + 13*bp(120)/150 + bp(200)/180 + bp(202)/18 + 13*bp(240)/150 + bp(300)/120 + bp(400)/180 + bp(404)/18 + bp(600)/7200 + bp(604)/16 + bp(1200)/7200.

Links

Crossrefs

Cf. A338981 (unoriented), A338982 (chiral), A338983 (achiral), A338964 (up to n colors), A337895 (5-cell), A337952 (8-cell vertices, 16-cell facets), A337956 (16-cell vertices, 8-cell facets), A338948 (24-cell).

Programs

  • Mathematica
    bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
CoefficientList[2bp[4]/15+bp[6]/5+2bp[8]/15+bp[10]/6+7bp[12]/150+bp[16]/25+bp[20]/180+bp[22]/18+7bp[24]/150+bp[30]/120+bp[32]/25+bp[40]/180+bp[44]/18+bp[60]/7200+bp[62]/16+bp[120]/7200,x]

Formula

A338964(n) = Sum_{j=1..Min(n,120)} a(n) * binomial(n,j).
a(n) = A338981(n) + A338982(n) = 2*A338981(n) - A338983(n) = 2*A338982(n) + A338983(n).
G.f.: 2*bp(4)/15 + bp(6)/5 + 2*bp(8)/15 + bp(10)/6 + 7*bp(12)/150 + bp(16)/25 + bp(20)/180 + bp(22)/18 + 7*bp(24)/150 + bp(30)/120 + bp(32)/25 + bp(40)/180 + bp(44)/18 + bp(60)/7200 + bp(62)/16 + bp(120)/7200, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.

A338981 Number of unoriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

Original entry on oeis.org

0, 1, 92307499707443390526727850063502, 124792381938502167392061689732085833655832902312754962, 122697712831831745940423467267565845711242845618544066030140191642464
Offset: 0

Views

Author

Robert A. Russell, Dec 13 2020

Keywords

Comments

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. For n>120, a(n) = 0.
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide generating functions here using bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k.
For the 600 facets of the 600-cell (vertices of the 120-cell), the generating function is bp(20)/15 + bp(30)/10 + bp(40)/15 + bp(50)/12 + 43*bp(60)/300 + bp(66)/10 + bp(100)/360 + bp(104)/9 + bp(114)/12 + 13*bp(120)/300 + bp(150)/240 + bp(152)/8 + bp(200)/360 + bp(208)/36 + 61*bp(300)/14400 + bp(302)/32 + bp(330)/240 + bp(600)/14400.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is bp(24)/15 + bp(36)/10 + bp(48)/15 + bp(60)/12 + 7*bp(72)/300 + 3*bp(76)/25 + bp(84)/10 + 41*bp(120)/360 + bp(132)/12 + 7*bp(144)/300 + bp(152)/50 + bp(180)/240 + bp(182)/8 + 11*bp(240)/360 + 61*bp(360)/14400 + bp(364)/32 + bp(396)/240 + bp(720)/14400.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is bp(40)/15 + bp(60)/10 + bp(80)/15 + bp(100)/12 + 43*bp(120)/300 + bp(128)/10 + bp(200)/360 + bp(202)/9 + bp(216)/12 + 13*bp(240)/300 + bp(300)/240 + bp(302)/8 + bp(400)/360 + bp(404)/36 + 61*bp(600)/14400 + bp(604)/32 + bp(640)/240 + bp(1200)/14400.

Links

Crossrefs

Cf. A338980 (oriented), A338982 (chiral), A338983 (achiral), A338965 (up to n colors), A000389 (5-cell), A128767 (8-cell vertices, 16-cell facets), A337957 (16-cell vertices, 8-cell facets), A338949 (24-cell).

Programs

  • Mathematica
    bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
CoefficientList[bp[4]/15+bp[6]/10+bp[8]/15+bp[10]/12+7bp[12]/300+bp[16]/50+bp[17]/10+bp[19]/10+bp[20]/360+bp[22]/36+bp[23]/12+7bp[24]/300+bp[27]/12+bp[30]/240+bp[31]/8+bp[32]/50+bp[40]/360+bp[44]/36+bp[60]/14400+bp[61]/240+bp[62]/32+bp[75]/240+bp[120]/14400,x]

Formula

A338965(n) = Sum_{j=1..Min(n,120)} a(n) * binomial(n,j).
a(n) = A338980(n) - A338982(n) = (A338980(n) + A338983(n)) / 2 = A338982(n) + A338983(n).
G.f.: bp(4)/15 + bp(6)/10 + bp(8)/15 + bp(10)/12 + 7bp(12)/300 + bp(16)/50 + bp(17)/10 + bp(19)/10 + bp(20)/360 + bp(22)/36 + bp(23)/12 + 7bp(24)/300 + bp(27)/12 + bp(30)/240 + bp(31)/8 + bp(32)/50 + bp(40)/360 + bp(44)/36 + bp(60)/14400 + bp(61)/240 + bp(62)/32 + bp(75)/240 + bp(120)/14400, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.

A338983 Number of chiral pairs of colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

Original entry on oeis.org

0, 1, 314843647550280564734, 5068890957389326592282175518285751, 11893730796581701705423717900461048616681772, 220581437248293418784474364671733389683204494492535
Offset: 0

Views

Author

Robert A. Russell, Dec 13 2020

Keywords

Comments

An achiral coloring is identical to its reflection. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. For n>75, a(n) = 0.
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide generating functions here using bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k.
For the 600 facets of the 600-cell (vertices of the 120-cell), the generating function is bp(60)/5 + bp(66)/5 + bp(104)/6 + bp(114)/6 + bp(152)/4 + bp(300)/120 + bp(330)/120.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is bp(76)/5 + bp(84)/5 + bp(120)/6 + bp(132)/6 + bp(182)/4 + bp(360)/120 + bp(396)/120.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is bp(120)/5 + bp(128)/5 + bp(202)/6 + bp(216)/6 + bp(302)/4 + bp(600)/120 + bp(640)/120.

Links

Crossrefs

Cf. A338980 (oriented), A338981 (unoriented), A338982 (chiral), A338967 (up to n colors), A132366 (5-cell), A337955 (8-cell vertices, 16-cell facets), A337958 (16-cell vertices, 8-cell facets), A338951 (24-cell).

Programs

  • Mathematica
    bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
CoefficientList[bp[17]/5+bp[19]/5+bp[23]/6+bp[27]/6+bp[31]/4+bp[61]/120+bp[75]/120,x]

Formula

A338967(n) = Sum_{j=1..Min(n,75)} a(n) * binomial(n,j).
a(n) = 2*A338981(n) - A338980(n) = A338980(n) - 2*A338982(n) = A338981(n) - A338982(n).
G.f.: bp(17)/5 + bp(19)/5 + bp(23)/6 + bp(27)/6 + bp(31)/4 + bp(61)/120 + bp(75)/120, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.

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 1-5 of 5 results.

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