A331360 Number of chiral pairs of colorings of the edges of a tesseract with n available colors.

0, 11158298, 4825452718593, 48038354542204960, 60632976384183154375, 20725679827848535509690, 2876113731787568888218778, 206323339833986421110604288, 8941884948181243949620880070

Offset

1,2

Comments

A tesseract is a regular 4-dimensional orthotope or hypercube with 16 vertices and 32 edges. Its Schläfli symbol is {4,3,3}. The chiral colorings of its edges come in pairs, each the reflection of the other. Also the number of chiral pairs of colorings of the triangular faces of a regular 4-dimensional orthoplex {3,3,4} with n available colors.

Links

Table of n, a(n) for n=1..9.

Index entries for linear recurrences with constant coefficients, signature (33, -528, 5456, -40920, 237336, -1107568, 4272048, -13884156, 38567100, -92561040, 193536720, -354817320, 573166440, -818809200, 1037158320, -1166803110, 1166803110, -1037158320, 818809200, -573166440, 354817320, -193536720, 92561040, -38567100, 13884156, -4272048, 1107568, -237336, 40920, -5456, 528, -33, 1).

Formula

a(n) = (48*n^4 - 92*n^8 + 32*n^12 + 3*n^16 + 24*n^18 - 16*n^20 + n^32) / 384.

a(n) = 11158298*C(n,2) + 4825419243699*C(n,3) + 48019052798280376*C(n,4) + 60392832865887732525*C(n,5) + 20362602448352682660450*C(n,6) + 2732305584323178619545720*C(n,7) + 183891356930602707657018720*C(n,8) + 7186781660616776435004792900*C(n,9) + 179941570948806294173832581400*C(n,10) + 3092495918794375534919002047600*C(n,11) + 38355721930663201428803366004000*C(n,12) + 356388702642050543223746618030400*C(n,13) + 2552262270629803579790727658473600*C(n,14) + 14398742619650630430045069333120000*C(n,15) + 65081946248235477116326789514496000*C(n,16) + 238774230958640305192143667115328000*C(n,17) + 718111905257279415879360961204608000*C(n,18) + 1783226074397879200641306482407680000*C(n,19) + 3674025240535453233675992278371840000*C(n,20) + 6297428247692138525542940292326400000*C(n,21) + 8984640042458034573900227275929600000*C(n,22) + 10651431202956156039912718487654400000*C(n,23) + 10448264801973961157855568414105600000*C(n,24) + 8418935641672774875938561280000000000*C(n,25) + 5510766716064148076659382317056000000*C(n,26) + 2882400456553496466714071801856000000*C(n,27) + 1175640370514915165746352603136000000*C(n,28) + 360177463966855890088916582400000000*C(n,29) + 77945658076061560043023564800000000*C(n,30) + 10621166594979816972895518720000000*C(n,31) + 685236554514826901477130240000000*C(n,32), where the coefficient of C(n,k) is the number of colorings using exactly k colors.

a(n) = A331358(n) - A331359(n) = (A331358(n) - A331361(n)) / 2 = A331359(n) - A331361(n).

Mathematica

Table[(48n^4 - 92n^8 + 32n^12 + 3n^16 + 24n^18 - 16n^20 + n^32)/384, {n, 1, 25}]

Crossrefs

Cf. A331358 (oriented), A331359 (unoriented), A331361 (achiral).

Cf. A331352 (simplex), A331356 (orthoplex), A338954 (24-cell), A338966 (120-cell, 600-cell).

Sequence in context: A075093 A206519 A270491 * A232445 A187375 A219749

Adjacent sequences: A331357 A331358 A331359 * A331361 A331362 A331363

Keyword

nonn,easy

Author

Robert A. Russell, Jan 14 2020

Status

approved