A071397 -id:A071397 - 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

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)).

A071398 Rounded total surface area of a regular icosahedron with edge length n.

Original entry on oeis.org

0, 9, 35, 78, 139, 217, 312, 424, 554, 701, 866, 1048, 1247, 1464, 1697, 1949, 2217, 2503, 2806, 3126, 3464, 3819, 4192, 4581, 4988, 5413, 5854, 6313, 6790, 7283, 7794, 8323, 8868, 9431, 10011, 10609, 11224, 11856, 12505, 13172, 13856, 14558, 15277

Offset: 0

Rick L. Shepherd, May 29 2002

			a(4)=139 because round(5*4^2*sqrt(3)) = round(80*1.73205...) = round(138.56...) = 139.

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
Eric Weisstein's World of Mathematics, Platonic Solid

Cf. A070169 (tetrahedron), A033581 (cube), A071396 (octahedron), A071397 (dodecahedron), A071402 (volume of icosahedron).

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

With[{c=5Sqrt[3]},Round[c Range[0,50]^2]] (* Harvey P. Dale, May 20 2011 *)

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

from math import isqrt
def A071398(n): return (m:=isqrt(k:=75*n**4))+int(k>m*(m+1)) # Chai Wah Wu, Jun 05 2025

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

A071401 Rounded volume of a regular dodecahedron with edge length n.

Original entry on oeis.org

0, 8, 61, 207, 490, 958, 1655, 2628, 3924, 5586, 7663, 10200, 13242, 16836, 21028, 25863, 31388, 37649, 44691, 52561, 61305, 70968, 81597, 93237, 105935, 119736, 134687, 150833, 168221, 186896, 206904, 228292, 251105, 275390, 301191, 328556

Offset: 0

Rick L. Shepherd, May 29 2002

			a(6)=1665 because round(6^3*(15+7*sqrt(5))/4)=round(216*7.6631...)=round(1655.23...)=1665.

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

Eric Weisstein's World of Mathematics, Dodecahedron

Cf. A000578 (cube), A071399 (tetrahedron), A071400 (octahedron), A071402 (icosahedron), A071397 (total surface area of dodecahedron).

Table[Floor[n^3 (15+7Sqrt[5])/4+1/2],{n,0,50}]  (* Harvey P. Dale, Apr 25 2011 *)

for(n=0,100,print1(round(n^3*(15+7*sqrt(5))/4),","))

a(n) = round(n^3 * (15+7*sqrt(5))/4)

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

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

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A071398 Rounded total surface area of a regular icosahedron with edge length n.

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A071401 Rounded volume of a regular dodecahedron with edge length n.

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