A274902 -id:A274902 - cp's OEIS frontend

A274900 Number of (not necessarily proper) vertex colorings of the truncated cube using at most n colors.

1, 352744, 5884691769, 5864100125056, 1241764261950625, 98716288267057896, 3991275742289356969, 98382635628154476544, 1661800900370941653561, 20833333346104183585000, 205202764127643987528241, 1656184316900213910466944, 11308349383297867766174569

Offset: 1

Marko Riedel, Jul 10 2016

Also the number of vertex colorings of the rhombicuboctahedron up to rotation and reflection. - Peter Kagey, Nov 27 2024

			Cycle index: 1/48*s[1]^24 + 1/8*s[2]^10*s[1]^4 + 13/48*s[2]^12 + 1/6*s[3]^8 + 1/4*s[4]^6 + 1/6*s[6]^4.

Marko R. Riedel et al., Truncated objects coloring, Mathematics Stack Exchange (Jul 10 2016).
Marko R. Riedel, Maple code for sequences A274900, A274901, A274902.
Wikipedia, Truncated cube

Cf. A274901, A274902.

[1/48*n^24+1/8*n^14+13/48*n^12+1/6*n^8+1/4*n^6+1/6*n^4: n in [1..20]]; // Vincenzo Librandi, Jul 11 2016

Table[1/48 n^24 + 1/8 n^14 + 13/48 n^12 + 1/6 n^8 + 1/4 n^6 + 1/6 n^4, {n, 25}] (* Vincenzo Librandi, Jul 11 2016 *)

a(n) = 1/48*n^24 + 1/8*n^14 + 13/48*n^12 + 1/6*n^8 + 1/4*n^6 + 1/6*n^4 \\ Felix Fröhlich, Jul 12 2016

a(n) = 1/48*n^24 + 1/8*n^14 + 13/48*n^12 + 1/6*n^8 + 1/4*n^6 + 1/6*n^4 = n^4*(n^20 + 6*n^10 + 13*n^8 + 8*n^4 + 12*n^2 + 8)/48.

A274901 Number of (not necessarily proper) face colorings of the truncated cube using at most n colors.

Original entry on oeis.org

1, 554, 109152, 5747200, 128538250, 1640929626, 14167981324, 91769978112, 477063389475, 2084653722250, 7914860972876, 26756396132544, 82046630783572, 231537699283450, 608260629969000, 1501341920229376, 3508131297671589, 7809071314434282, 16646760371737000

Offset: 1

Marko Riedel, Jul 10 2016

			Cycle index: 1/48*s[1]^14 + 1/8*s[1]^6*s[2]^4 + 1/16*s[2]^5*s[1]^4 + 1/16*s[2]^6*s[1]^2 + 7/48*s[2]^7 + 1/6*s[1]^2*s[3]^4 + 1/8*s[4]^3*s[1]^2 + 1/8*s[4]^3*s[2] + 1/6*s[6]^2*s[2].

Marko R. Riedel et al., Truncated objects coloring, Mathematics Stack Exchange (Jul 10 2016).
Wikipedia, Truncated cube

Cf. A274900, A274902.

[1/48*n^14+1/8*n^10+1/16*n^9+1/16*n^8+7/48*n^7+1/6*n^6+1/8*n^5+ 1/8*n^4+1/6*n^3: n in [1..20]]; // Vincenzo Librandi, Jul 11 2016

Table[1/48 n^14 + 1/8 n^10 + 1/16 n^9 + 1/16 n^8 + 7/48 n^7 + 1/6 n^6 + 1/8 n^5 + 1/8 n^4 + 1/6 n^3, {n, 25}] (* Vincenzo Librandi, Jul 11 2016 *)

a(n) = 1/48*n^14 + 1/8*n^10 + 1/16*n^9 + 1/16*n^8 + 7/48*n^7 + 1/6*n^6 + 1/8*n^5 + 1/8*n^4 + 1/6*n^3 = n^3*(n + 1)*(n^10 - n^9 + n^8 - n^7 + 7*n^6 - 4*n^5 + 7*n^4 + 8*n^2 - 2*n + 8)/48.

cp's OEIS Frontend

A274900 Number of (not necessarily proper) vertex colorings of the truncated cube using at most n colors.

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