A319587 The number of distinct solid nets of the six convex regular 4D-polytopes in the order of their 3D-cell count.

3, 261, 110912, 17895697067018274

Offset: 1

Author

Frank M Jackson, Sep 23 2018

Keywords

• nonn
• fini

Comments

These values have been taken from the Buekenhout (1998) paper (see link). During the unfolding of these solid nets along their common face, the possibility of any overlapping is ignored.

This finite sequence is fully determined but a(5) and a(6) are too large to be displayed in data. See formulas below to calculate these terms.

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..6
• F. Buekenhout and M. Parker, The number of nets of the regular convex polytopes in dimension >= 4, Discrete Mathematics 186 (1998) 69-94.

Crossrefs

Cf. A091159, A201187.

Programs

Mathematica

{3, (82944+12*16+24*8+4*2304+6*128+12*96+12*192+12*288)/(2^7*3), 2^5(2^7*3^3+1+3^2), 6(2^19*5688888889+347), 2^7*5^2*7^3(2^114*3^78*5^20*7^33+2^47*3^18*5^2*7^12*53^5*2311^3+239^2*3931^2), 2^188*3^102*5^20*7^36*11^48*23^48*29^30}

Formula

a(1) = 3;

a(2) = (82944 + 12*16 + 24*8 + 4*2304 + 6*128 + 12*96 + 12*192 + 12*288)/(2^7 * 3) = 261;

a(3) = 2^5*(2^7 * 3^3 + 1 + 3^2) = 110912;

a(4) = 6*(2^19 * 5688888889 + 347) = 17895697067018274;

a(5) = 2^7 * 5^2 * 7^3 * (2^114 * 3^78 * 5^20 * 7^33 + 2^47 * 3^18 * 5^2 * 7^12 * 53^5 * 2311^3 + 239^2 * 3931^2);

a(6) = 2^188 * 3^102 * 5^20 * 7^36 * 11^48 * 23^48 * 29^30.