A002194 -id:A002194 - cp's OEIS frontend

A194416 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) = 0, where r=sqrt(3) and < > denotes fractional part.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 63, 66, 69, 78, 81, 84, 93, 96, 99, 108, 111, 123, 126, 138, 141, 153, 156, 159, 162, 165, 168, 171, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 204, 207, 216, 219, 222, 231, 234

Offset: 1

Clark Kimberling, Aug 24 2011

nonn

Every term is divisible by 3; see A194368.

Cf. A002194, A194368, A194415, A194417, A194418.

r = Sqrt[3]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 150}];
Flatten[Position[t1, 1]]           (* A194415 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]           (* A194416 *)
%/3                                (* A194417 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 500}];
Flatten[Position[t3, 1]]           (* A194418 *)

A214726 Decimal expansion of the perimeter of Cairo and Prismatic tiles.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Jul 26 2012

An algebraic integer with degree 4 and minimal polynomial x^4 - 16x^2 + 16. - Charles R Greathouse IV, Apr 21 2016

Length of the longest diagonal in a regular 12-gon with unit side. - Mohammed Yaseen, Dec 13 2020

			3.8637033051562731469989727989....

Paolo Xausa, Table of n, a(n) for n = 1..10000
Ping Ngai Chung, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, and Elena Wikner, Isoperimetric Pentagonal Tilings, Not. AMS, Vol. 59, No. 5 (May 2012) pp. 632-640; arXiv preprint, arXiv:1111.6161 [math.MG], 2011.

Cf. A002193, A002194, A019884.

First[RealDigits[Csc[Pi/12],10,100]] (* Paolo Xausa, Oct 19 2023 *)

2*sqrt(2+sqrt(3)) \\ Charles R Greathouse IV, Apr 21 2016

Equals 2*(sqrt(2+sqrt(3))).
Equals csc(Pi/12). - Amiram Eldar, May 28 2021
Equals sqrt(2) + sqrt(6). - Vaclav Kotesovec, May 28 2021
Equals Product_{k>=1} (25 - 144*k^2)/(100 - 144*k^2). - Antonio Graciá Llorente, Jul 13 2024
Equals 4 * A019884. - Alois P. Heinz, Jul 14 2024

a(100) corrected by Georg Fischer, Jul 12 2021

A377274 Decimal expansion of the surface area of a truncated tetrahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 23 2024

			12.12435565298214105469212439054110656859963677667...

Eric Weisstein's World of Mathematics, Truncated Tetrahedron.
Wikipedia, Truncated tetrahedron.

Cf. A377275 (volume), A377276 (circumradius), A093577 (midradius), A377277 (Dehn invariant).

Cf. A002194 (analogous for a regular tetrahedron).

First[RealDigits[7*Sqrt[3], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedTetrahedron", "SurfaceArea"], 10, 100]]

Equals 7*sqrt(3) = 7*A002194.

A377996 Decimal expansion of the dihedral angle, in radians, between triangular and square faces in a (small) rhombicosidodecahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 15 2024

Also the dihedral angle, in radians, between square and hexagonal faces in a truncated icosidodecahedron (great rhombicosidodecahedron).

			2.7767288254763100567352450533728425364989164423512...

Eric Weisstein's World of Mathematics, Great Rhombicosidodecahedron.
Eric Weisstein's World of Mathematics, Small Rhombicosidodecahedron.

Cf. A002194, A010472, A377995.

First[RealDigits[ArcCos[-(Sqrt[3] + Sqrt[15])/6], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["Rhombicosidodecahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-(sqrt(3) + sqrt(15))/6) = arccos(-(A002194 + A010472)/6).

A384286 Decimal expansion of the surface area of a gyroelongated pentagonal rotunda with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, May 30 2025

The gyroelongated pentagonal rotunda is Johnson solid J_25.

			31.00745430323851474443564586571797490853203978248...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyroelongated pentagonal rotunda.

Cf. A384285 (volume).

Cf. A002163, A002194, A179553, A179591, A179640, A384141, A384284.

First[RealDigits[(15*Sqrt[3] + Sqrt[650 + 290*Sqrt[5]])/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J25", "SurfaceArea"], 10, 100]]

Equals (15*sqrt(3) + sqrt(650 + 290*sqrt(5)))/2 = (15*A002194 + sqrt(650 + 290*A002163))/2.

Equals the largest root of 256*x^8 - 339200*x^6 + 98924000*x^4 - 9264250000*x^2 + 176295015625.

A385506 Decimal expansion of the volume of a triaugmented triangular prism with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 01 2025

The triaugmented triangular prism is Johnson solid J_51.

			1.140119483078766847782705947481317131020537251141...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Triaugmented triangular prism.

Cf. A097715 (surface area).

Cf. A002194, A010466, A010503, A120011, A385505.

First[RealDigits[(Sqrt[8] + Sqrt[3])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J51", "Volume"], 10, 100]]

Equals 1/sqrt(2) + sqrt(3)/4 = A010503 + A120011 = (A010466 + A002194)/4.

Equals the largest root of 256*x^4 - 352*x^2 + 25.

A385803 Decimal expansion of the surface area of a parabiaugmented dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 09 2025

The parabiaugmented dodecahedron is Johnson solid J_59.

Also the surface area of a metabiaugmented dodecahedron (Johnson solid J_60) with unit edge.

			21.5349010248118624614087356276507769114307548346...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Parabiaugmented dodecahedron.

Cf. A385802 (volume).

Cf. A002194, A010476, A385696, A385805.

First[RealDigits[5/2*(Sqrt[3] + Sqrt[25 + 10*Sqrt[5]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J59", "SurfaceArea"], 10, 100]]

Equals (5/2)*(sqrt(3) + sqrt(5*(5 + 2*sqrt(5)))) = (5/2)*(A002194 + sqrt(5*(5 + A010476))).

Equals the largest root of x^8 - 700*x^6 + 121250*x^4 - 5421875*x^2 + 390625.

A385805 Decimal expansion of the surface area of a triaugmented dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 09 2025

The triaugmented dodecahedron is Johnson solid J_61.

			21.97948713368399215555903157714450777070188723...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Triaugmented dodecahedron.

Cf. A385804 (volume).

Cf. A002194, A010476, A385696, A385803.

First[RealDigits[3/4*(5*Sqrt[3] + 3*Sqrt[25 + 10*Sqrt[5]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J61", "SurfaceArea"], 10, 100]]

Equals (3/4)*(5*sqrt(3) + 3*sqrt(5*(5 + 2*sqrt(5)))) = (3/4)*(5*A002194 + 3*sqrt(5*(5 + A010476))).

Equals the largest root of 256*x^8 - 172800*x^6 + 26244000*x^4 - 1230187500*x^2 + 8303765625.

A004549 Expansion of sqrt(3) in base 4.

Original entry on oeis.org

1, 2, 3, 2, 3, 1, 2, 1, 3, 2, 2, 3, 2, 2, 0, 1, 1, 2, 0, 1, 0, 3, 0, 2, 2, 2, 2, 1, 3, 0, 3, 2, 3, 0, 2, 1, 1, 1, 3, 1, 0, 0, 2, 3, 1, 1, 3, 0, 0, 1, 3, 2, 0, 2, 3, 2, 0, 0, 3, 2, 3, 2, 0, 2, 1, 0, 2, 1, 1, 3, 1, 2, 0, 0, 3, 1, 0, 3, 0, 3, 0, 1, 1, 0, 3, 3, 1, 2, 2, 1, 0, 1, 3, 2, 1, 2, 0, 3, 0

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A002194, A004547, A004548,

Cf. A004540, A004559, A004575.

Prune(Reverse(IntegerToSequence(Isqrt(3*4^200), 4))); // Vincenzo Librandi, Jan 07 2018

RealDigits[Sqrt[3], 4, 130][[1]] (* Vincenzo Librandi, Jan 07 2018 *)

A119344 Integer lengths of Theodorus-primes: numbers n such that the concatenation of the first n decimal digits of the Theodorus's constant sqrt(3) is prime.

Original entry on oeis.org

2, 3, 19, 111, 116, 641, 5411, 170657

Offset: 1

Eric W. Weisstein, May 15 2006

			sqrt(3) = 1.732050807568877..., so
a(1) = 2 (17 with 2 decimal digits is the 1st prime in the decimal expansion),
a(2) = 3 (173 with 3 decimal digits is the 2nd prime in the decimal expansion).

Eric Weisstein's World of Mathematics, Constant Primes
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Theodorus's Constant Digits

Cf. A119343 (Theodorus-primes).

Cf. A002194 (decimal expansion of sqrt(3)).

nn = 1000; digs = RealDigits[Sqrt[3], 10, nn][[1]]; n = 0; t = {}; Do[n = 10*n + digs[[d]]; If[PrimeQ[n], AppendTo[t, d]], {d, nn}]; t (* T. D. Noe, Dec 05 2011 *)
Module[{nn=171000,c},c=RealDigits[Sqrt[3],10,nn][[1]];Select[Range[ nn], PrimeQ[ FromDigits[Take[c,#]]]&]] (* Harvey P. Dale, May 13 2017 *)

Edited by Charles R Greathouse IV, Apr 27 2010

a(8) = 170657 from Eric W. Weisstein, Aug 18 2013

cp's OEIS Frontend

A194416 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) = 0, where r=sqrt(3) and < > denotes fractional part.

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A214726 Decimal expansion of the perimeter of Cairo and Prismatic tiles.

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A377274 Decimal expansion of the surface area of a truncated tetrahedron with unit edge length.

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A377996 Decimal expansion of the dihedral angle, in radians, between triangular and square faces in a (small) rhombicosidodecahedron.

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A384286 Decimal expansion of the surface area of a gyroelongated pentagonal rotunda with unit edge.

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A385506 Decimal expansion of the volume of a triaugmented triangular prism with unit edge.

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A385803 Decimal expansion of the surface area of a parabiaugmented dodecahedron with unit edge.

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A385805 Decimal expansion of the surface area of a triaugmented dodecahedron with unit edge.

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