A070169 -id:A070169 - cp's OEIS frontend

A033581 a(n) = 6*n^2.

0, 6, 24, 54, 96, 150, 216, 294, 384, 486, 600, 726, 864, 1014, 1176, 1350, 1536, 1734, 1944, 2166, 2400, 2646, 2904, 3174, 3456, 3750, 4056, 4374, 4704, 5046, 5400, 5766, 6144, 6534, 6936, 7350, 7776, 8214, 8664, 9126, 9600, 10086, 10584, 11094, 11616

Offset: 0

N. J. A. Sloane

Number of edges of a complete 4-partite graph of order 4n, K_n,n,n,n. - Roberto E. Martinez II, Oct 18 2001
Number of edges of the complete bipartite graph of order 7n, K_n, 6n. - Roberto E. Martinez II, Jan 07 2002
Number of edges in the line graph of the product of two cycle graphs, each of order n, L(C_n x C_n). - Roberto E. Martinez II, Jan 07 2002
Total surface area of a cube of edge length n. See A000578 for cube volume. See A070169 and A071399 for surface area and volume of a regular tetrahedron and links for the other Platonic solids. - Rick L. Shepherd, Apr 24 2002
a(n) can represented as n concentric hexagons (see example). - Omar E. Pol, Aug 21 2011
Sequence found by reading the line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A003154 in the same spiral. - Omar E. Pol, Sep 08 2011
Together with 1, numbers m such that floor(2*m/3) and floor(3*m/2) are both squares. Example: floor(2*150/3) = 100 and floor(3*150/2) = 225 are both squares, so 150 is in the sequence. - Bruno Berselli, Sep 15 2014
a(n+1) gives the number of vertices in a hexagon-like honeycomb built from A003215(n) congruent regular hexagons (see link). Example: a hexagon-like honeycomb consisting of 7 congruent regular hexagons has 1 core hexagon inside a perimeter of six hexagons. The perimeter has 18 vertices. The core hexagon has 6 vertices. a(2) = 18 + 6 = 24 is the total number of vertices. - Ivan N. Ianakiev, Mar 11 2015
a(n) is the area of the Pythagorean triangle whose sides are (3n, 4n, 5n). - Sergey Pavlov, Mar 31 2017
More generally, if k >= 5 we have that the sequence whose formula is a(n) = (2*k - 4)*n^2 is also the sequence found by reading the line from 0, in the direction 0, (2*k - 4), ..., in the square spiral whose vertices are the generalized k-gonal numbers. In this case k = 5. - Omar E. Pol, May 13 2018
The sequence also gives the number of size=1 triangles within a match-made hexagon of size n. - John King, Mar 31 2019
For hexagons, the number of matches required is A045945; thus number of size=1 triangles is A033581; number of larger triangles is A307253 and total number of triangles is A045949. See A045943 for analogs for Triangles; see A045946 for analogs for Stars. - John King, Apr 04 2019

			From _Omar E. Pol_, Aug 21 2011: (Start)
Illustration of initial terms as concentric hexagons:
.
.                                 o o o o o o
.                                o           o
.              o o o o          o   o o o o   o
.             o       o        o   o       o   o
.   o o      o   o o   o      o   o   o o   o   o
.  o   o    o   o   o   o    o   o   o   o   o   o
.   o o      o   o o   o      o   o   o o   o   o
.             o       o        o   o       o   o
.              o o o o          o   o o o o   o
.                                o           o
.                                 o o o o o o
.
.    6            24                   54
.
(End)

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Bruno Berselli, An interpretation of initial terms.
Ivan N. Ianakiev, Hexagon-like honeycomb built form regular congruent hexagons.
Leo Tavares, Illustration: Diamond Star Rays
Eric Weisstein's World of Mathematics, Platonic Solid.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A032528. Central column of triangle A001283.
Cf. A000217, A000290, A001105, A009111, A033428, A033583, A085250, A086729, A152734, A152751.
Cf. A000578, A001318, A003215, A005897, A045943, A045945, A045949, A070169, A071399.
Cf. A017593 (first differences).

a033581 = (* 6) . (^ 2)  -- Reinhard Zumkeller, Apr 27 2014

seq(6*n^2,n=0..44); # Nathaniel Johnston, Jun 26 2011

6 Range[44]^2 (* Michael De Vlieger, Apr 02 2017 *)
LinearRecurrence[{3,-3,1},{0,6,24},50] (* Harvey P. Dale, Jul 03 2017 *)

vector(100,n,6*(n-1)^2) \\ Derek Orr, Mar 11 2015

a(n) = A000290(n)*6. - Omar E. Pol, Dec 11 2008
a(n) = A001105(n)*3 = A033428(n)*2. - Omar E. Pol, Dec 13 2008
a(n) = 12*n + a(n-1) - 6, with a(0)=0. - Vincenzo Librandi, Aug 05 2010
G.f.: 6*x*(1+x)/(1-x)^3. - Colin Barker, Feb 14 2012
For n > 0: a(n) = A005897(n) - 2. - Reinhard Zumkeller, Apr 27 2014
a(n) = 3*floor(1/(1-cos(1/n))) = floor(1/(1-n*sin(1/n))) for n > 0. - Clark Kimberling, Oct 08 2014
a(n) = t(4*n) - 4*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(4*n) - 4*A000217(n). - Bruno Berselli, Aug 31 2017
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/36.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/72 (A086729).
Product_{n>=1} (1 + 1/a(n)) = sqrt(6)*sinh(Pi/sqrt(6))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(6)*sin(Pi/sqrt(6))/Pi. (End)
E.g.f.: 6*exp(x)*x*(1 + x). - Stefano Spezia, Aug 19 2022

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2001

A171972 Greatest integer k such that k/n^2 < sqrt(3).

Original entry on oeis.org

0, 1, 6, 15, 27, 43, 62, 84, 110, 140, 173, 209, 249, 292, 339, 389, 443, 500, 561, 625, 692, 763, 838, 916, 997, 1082, 1170, 1262, 1357, 1456, 1558, 1664, 1773, 1886, 2002, 2121, 2244, 2371, 2501, 2634, 2771, 2911, 3055, 3202, 3353, 3507, 3665, 3826, 3990, 4158

Offset: 0

Reinhard Zumkeller, Jan 20 2010

nonn

Integer part of the surface area of a regular tetrahedron with edge length n.
A171970(n)*A005843(n) <= a(n);
a(n) <= 4*A171971(n); 0 <= a(n) - 4*A171971(n) < 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Tetrahedron
Wikipedia, Tetrahedron

Cf. A002194, A070169, A171974, A171973, A171975, A000290, A293410.

a171972 = floor . (* sqrt 3) . fromInteger . a000290
-- Reinhard Zumkeller, Dec 15 2012

z = 120; r = Sqrt[3];
Table[Floor[r*n^2], {n, 0, z}]; (* A171972 *)
Table[Ceiling[r*n^2], {n, 0, z}]; (* A293410 *)
Table[Round[r*n^2], {n, 0, z}]; (* A070169. -  Clark Kimberling, Oct 11 2017 *)

a(n) = floor(n^2 * sqrt(3)).
a(n) = A022838(n^2);
a(n) = A293410(n) - 1 for n > 0.

A071399 Rounded volume of a regular tetrahedron with edge length n.

Original entry on oeis.org

0, 0, 1, 3, 8, 15, 25, 40, 60, 86, 118, 157, 204, 259, 323, 398, 483, 579, 687, 808, 943, 1091, 1255, 1434, 1629, 1841, 2071, 2320, 2587, 2874, 3182, 3511, 3862, 4235, 4632, 5053, 5498, 5970, 6467, 6991, 7542, 8122, 8731, 9370, 10039, 10739, 11471, 12236

Offset: 0

Rick L. Shepherd, May 29 2002

			a(4)=8 because round(4^3*sqrt(2)/12)=round(64*.11785...)=round(7.542...)=8.

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

Eric Weisstein's World of Mathematics, Tetrahedron
Eric Weisstein's World of Mathematics, Platonic Solid

Cf. A000578 (cube), A071400 (octahedron), A071401 (dodecahedron), A071402 (icosahedron), A070169 (total surface area of tetrahedron).

With[{c=Sqrt[2]/12},Round[c*Range[0,50]^3]] (* Harvey P. Dale, Feb 25 2015 *)

for(n=0,100,print1(round(n^3*sqrt(2)/12),","))

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

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)).

A293410 Least integer k such that k/n^2 > sqrt(3).

Original entry on oeis.org

0, 2, 7, 16, 28, 44, 63, 85, 111, 141, 174, 210, 250, 293, 340, 390, 444, 501, 562, 626, 693, 764, 839, 917, 998, 1083, 1171, 1263, 1358, 1457, 1559, 1665, 1774, 1887, 2003, 2122, 2245, 2372, 2502, 2635, 2772, 2912, 3056, 3203, 3354, 3508, 3666, 3827, 3991

Offset: 0

Clark Kimberling, Oct 11 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A002194, A171972, A070169.

z = 120; r = Sqrt[3];
Table[Floor[r*n^2], {n, 0, z}];   (* A171972 *)
Table[Ceiling[r*n^2], {n, 0, z}]; (* A293410 *)
Table[Round[r*n^2], {n, 0, z}];   (* A070169 *)

from math import isqrt
def A293410(n): return 1+isqrt(3*n**4-1) if n else 0 # Chai Wah Wu, Jul 31 2022

a(n) = ceiling(r*n^2), where r = sqrt(3).

a(n) = A171972(n) + 1 for n > 0.

A178988 Decimal expansion of volume of golden tetrahedron.

Original entry on oeis.org

7, 5, 7, 5, 5, 2, 2, 1, 2, 8, 1, 0, 1, 1, 4, 9, 2, 9, 7, 6, 9, 2, 0, 8, 0, 5, 6, 3, 0, 6, 4, 4, 5, 8, 0, 9, 2, 7, 0, 3, 7, 5, 3, 2, 6, 1, 9, 3, 9, 2, 9, 2, 1, 4, 7, 5, 9, 1, 2, 9, 9, 2, 1, 3, 9, 5, 2, 4, 5, 6, 5, 1, 0, 6, 0, 2, 5, 9, 4, 9, 6, 8, 8, 5, 3, 3, 6, 9, 9, 2, 8, 4, 4, 4, 9, 8, 4, 2, 5, 6, 9

Offset: 2

Jonathan Vos Post, Jan 03 2011

Volume of tetrahedron with edges 1, phi, phi^2, phi^3, phi^4, phi^5 where phi is the golden ratio (1+sqrt(5))/2.

A152149 records more recent developments about side-golden and angle-golden triangles, both of which, like the golden rectangle, have generalizations that match continued fractions. There is a unique triangle which is both side-golden and angle-golden. Is there a comparable tetrahedron? - Clark Kimberling, Mar 31 2011

			75.7552212810...

Clark Kimberling, "A New Kind of Golden Triangle." In Applications of Fibonacci Numbers: Proceedings of the Fourth International Conference on Fibonacci Numbers and Their Applications,' Wake Forest University (Ed. G. E. Bergum, A. N. Philippou, and A. F. Horadam). Dordrecht, Netherlands: Kluwer, pp. 171-176, 1991.
Theoni Pappas, "The Pentagon, the Pentagram & the Golden Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 188-189, 1989.

Marjorie Bicknell and Verner E. Hoggatt Jr., Golden Triangles, Rectangles, and Cuboids, Fib. Quart. 7, 73-91, 1969.
Frank M. Jackson and Eric W. Weisstein, Tetrahedron.
Clark Kimberling, A New Kind of Golden Triangle, In: Bergum G.E., Philippou A.N., Horadam A.F. (eds), Applications of Fibonacci Numbers. Springer, Dordrecht, pp. 171-176, 1991.
Robert Schoen, The Fibonacci Sequence in Successive Partitions of a Golden Triangle, Fib. Quart. 20, 159-163, 1982.
Eric W. Weisstein, Golden Triangle.

Cf. A001622, A071399, A171973, A070169, A126766, A010524, A093524, A093525, A093591, A152149.

RealDigits[Sqrt[275465/96 + 369575*Sqrt[5]/288], 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

sqrt(275465/96 + (369575*sqrt(5))/288) \\ Charles R Greathouse IV, May 27 2016

Equals sqrt(275465/96 + (369575*sqrt(5))/288).

The minimal polynomial is 20736*x^4 - 119000880*x^2 + 73225. - Joerg Arndt, Jul 25 2021

a(101) corrected by Georg Fischer, Jul 25 2021

cp's OEIS Frontend

A033581 a(n) = 6*n^2.

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A171972 Greatest integer k such that k/n^2 < sqrt(3).

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A071399 Rounded volume 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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A071397 Rounded total surface area of a regular dodecahedron 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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