A071398 -id:A071398 - cp's OEIS frontend

A070169 Rounded total surface area of a regular tetrahedron with edge length n.

0, 2, 7, 16, 28, 43, 62, 85, 111, 140, 173, 210, 249, 293, 339, 390, 443, 501, 561, 625, 693, 764, 838, 916, 998, 1083, 1171, 1263, 1358, 1457, 1559, 1665, 1774, 1886, 2002, 2122, 2245, 2371, 2501, 2634, 2771, 2912, 3055, 3203, 3353, 3507, 3665, 3826, 3991

Offset: 0

Rick L. Shepherd, Apr 24 2002

a(n) is the integer k that minimizes |k/n^2 - sqrt(3)|. - Clark Kimberling, Oct 11 2017

			a(3)=16 because round(3^2*sqrt(3)) = round(9*1.73205...) = round(15.5884...) = 16.

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, pp. 10-11.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Tetrahedron
Eric Weisstein's World of Mathematics, Platonic Solid

Cf. A033581 (cube), A071396 (octahedron), A071397 (dodecahedron), A071398 (icosahedron), A071399 (volume of tetrahedron).

[Round(n^2 * Sqrt(3)): n in [0..50]]; // Vincenzo Librandi, May 21 2011

Round[Sqrt[3]#]&/@(Range[0,50]^2) (* Harvey P. Dale, Sep 24 2012 *)

for(n=0,100,print1(round(n^2*sqrt(3)),","))

from math import isqrt
def A070169(n): return (m:=isqrt(k:=3*n**4))+(k-m*(m+1)>=1) # Chai Wah Wu, Jun 19 2024

a(n) = round(n^2 * sqrt(3)).

a(n) = A000194(3*n^4). - Christian Krause, Aug 04 2021; corrected by Chai Wah Wu, Jun 19 2024

A071402 Rounded volume of a regular icosahedron with edge length n.

Original entry on oeis.org

0, 2, 17, 59, 140, 273, 471, 748, 1117, 1590, 2182, 2904, 3770, 4793, 5987, 7363, 8936, 10719, 12724, 14964, 17454, 20205, 23231, 26545, 30160, 34089, 38345, 42942, 47893, 53209, 58906, 64995, 71490, 78404, 85749, 93540, 101789, 110509

Offset: 0

Rick L. Shepherd, May 29 2002

The printed reference given shows in a table on p. 10 that Volume is "2.18170a^3" (a is edge). Both PARI (see Example here) and a handheld calculator show that 2.18169 is correct for a 5-decimal-place approximation.

			a(6)=471 because round(6^3*(3 + sqrt(5))*5/12) = round(216*2.181694990...) = round(471.24...) = 471.

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, pp. 10-11.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Icosahedron

Cf. A000578 (cube), A071399 (tetrahedron), A071400 (octahedron), A071401 (dodecahedron), A071398 (total surface area of icosahedron).

Cf. A102208 ((3+Sqrt(5)) * 5/12).

[Round(n^3 * (3+Sqrt(5)) * 5/12): n in [0..50]]; // Vincenzo Librandi, May 21 2011

for(n=0,100,print1(round(n^3*(3+sqrt(5))*5/12),","))

a(n) = round(n^3 * (3+sqrt(5)) * 5/12).

A071396 Rounded total surface area of a regular octahedron with edge length n.

Original entry on oeis.org

0, 3, 14, 31, 55, 87, 125, 170, 222, 281, 346, 419, 499, 585, 679, 779, 887, 1001, 1122, 1251, 1386, 1528, 1677, 1833, 1995, 2165, 2342, 2525, 2716, 2913, 3118, 3329, 3547, 3772, 4005, 4244, 4489, 4742, 5002, 5269, 5543, 5823, 6111, 6405, 6707, 7015, 7330

Offset: 0

Rick L. Shepherd, May 23 2002

			a(3)=31 because round(2*3^2*sqrt(3)) = round(18*1.73205...) = round(31.1769...) = 31.

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, pp. 10-11.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Octahedron
Eric Weisstein's World of Mathematics, Platonic Solid

Cf. A070169 (tetrahedron), A033581 (cube), A071397 (dodecahedron), A071398 (icosahedron), A071400 (volume of octahedron).

[Round(2*n^2 * Sqrt(3)): n in [0..50]]; // Vincenzo Librandi, May 21 2011

Table[Round[2n^2 Sqrt[3]],{n,0,50}] (* Harvey P. Dale, Feb 19 2024 *)

for(n=0,100,print1(round(2*n^2*sqrt(3)),","))

from math import isqrt
def A071396(n): return (m:=isqrt(k:=3*n**4<<2))+int(k>m*(m+1)) # Chai Wah Wu, Jun 05 2025

a(n) = round(2 * n^2 * sqrt(3)).

A071397 Rounded total surface area of a regular dodecahedron with edge length n.

Original entry on oeis.org

0, 21, 83, 186, 330, 516, 743, 1012, 1321, 1672, 2065, 2498, 2973, 3489, 4047, 4645, 5285, 5967, 6689, 7453, 8258, 9105, 9993, 10922, 11892, 12904, 13957, 15051, 16186, 17363, 18581, 19841, 21141, 22483, 23866, 25291, 26757, 28264, 29812

Offset: 0

Rick L. Shepherd, May 28 2002

			a(4)=330 because round(3*4^2*sqrt(25 + 10*sqrt(5))) = round(48*6.88190...) = round(330.331...) = 330.

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, pp. 10-11.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Dodecahedron
Eric Weisstein's World of Mathematics, Platonic Solid

Cf. A070169 (tetrahedron), A033581 (cube), A071396 (octahedron), A071398 (icosahedron), A071401 (volume of dodecahedron).

[Round(3 * n^2 * Sqrt(25+10*Sqrt(5))): n in [0..50]]; // Vincenzo Librandi, May 21 2011

With[{c=3*Sqrt[25+10*Sqrt[5]]},Round[c*Range[0,40]^2]] (* Harvey P. Dale, Jul 06 2018 *)

for(n=0,100,print1(round(3*n^2*sqrt(25+10*sqrt(5))),","))

a(n) = round(3 * n^2 * sqrt(25 + 10*sqrt(5))).

cp's OEIS Frontend

A070169 Rounded total surface area of a regular tetrahedron with edge length n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A071402 Rounded volume of a regular icosahedron with edge length n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

PARI

Formula

A071396 Rounded total surface area of a regular octahedron with edge length n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A071397 Rounded total surface area of a regular dodecahedron with edge length n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula