A010524 -id:A010524 - cp's OEIS frontend

A377298 Decimal expansion of the surface area of a truncated cube with unit edge length.

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

Offset: 2

Paolo Xausa, Oct 25 2024

			32.4346643636148951726751573735281216767216730121...

Eric Weisstein's World of Mathematics, Truncated Cube.
Wikipedia, Truncated cube.

Cf. A377299 (volume), A294968 (circumradius), A010503 (midradius - 1), A377296 (Dehn invariant, negated).

Cf. A002193, A002194, A010469, A010524.

First[RealDigits[2*(6 + Sqrt[72] + Sqrt[3]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedCube", "SurfaceArea"], 10, 100]]

Equals 2*(6 + 6*sqrt(2) + sqrt(3)) = 2*(6 + 2*A002193 + A002194) = 12 + 2*A010524 + A010469.

A377345 Decimal expansion of the circumradius of a truncated cuboctahedron (great rhombicuboctahedron) with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 26 2024

			2.3176109128927665137791474633402948053450518945...

Eric Weisstein's World of Mathematics, Great Rhombicuboctahedron.
Wikipedia, Truncated cuboctahedron.

Cf. A377343 (surface area), A377344 (volume), A377346 (midradius).

Cf. A010524.

First[RealDigits[Sqrt[13 + 6*Sqrt[2]]/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedCuboctahedron", "Circumradius"], 10, 100]]

Equals sqrt(13 + 6*sqrt(2))/2 = sqrt(13 + A010524)/2.

A377346 Decimal expansion of the midradius of a truncated cuboctahedron (great rhombicuboctahedron) with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 26 2024

			2.26303343845371462359202580343253714222906720265...

Eric Weisstein's World of Mathematics, Great Rhombicuboctahedron.
Wikipedia, Truncated cuboctahedron.

Cf. A377343 (surface area), A377344 (volume), A377345 (circumradius).
Cf. A010527 (analogous for a cuboctahedron).
Cf. A010524, A230981.

First[RealDigits[Sqrt[3 + 3/Sqrt[2]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedCuboctahedron", "Midradius"], 10, 100]]

Equals sqrt(12 + 6*sqrt(2))/2 = sqrt(12 + A010524)/2 = sqrt(3 + 3/sqrt(2)) = sqrt(3 + A230981).

A041126 Numerators of continued fraction convergents to sqrt(72).

Original entry on oeis.org

8, 17, 280, 577, 9512, 19601, 323128, 665857, 10976840, 22619537, 372889432, 768398401, 12667263848, 26102926097, 430314081400, 886731088897, 14618011503752, 30122754096401, 496582077046168, 1023286908188737, 16869172608065960, 34761632124320657

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).

Cf. A010524, A041127.

Numerator[Convergents[Sqrt[72], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
a0[n_] := (-4+3*Sqrt[2])*(17+12*Sqrt[2])^n-((4+3*Sqrt[2])/(17+12*Sqrt[2])^n) // Simplify
a1[n_] := (1/(17+12*Sqrt[2])^n+(17+12*Sqrt[2])^n)/2 // FullSimplify
Flatten[MapIndexed[{a0[#], a1[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)

G.f.: -(x+1)*(x^2-9*x-8) / ((x^2-6*x+1)*(x^2+6*x+1)). - Colin Barker, Nov 05 2013
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a1(n),a0(n)] for n>0:
a0(n) = (-4+3*sqrt(2))*(17+12*sqrt(2))^n-((4+3*sqrt(2))/(17+12*sqrt(2))^n).
a1(n) = (1/(17+12*sqrt(2))^n+(17+12*sqrt(2))^n)/2. (End)

More terms from Colin Barker, Nov 05 2013

A041127 Denominators of continued fraction convergents to sqrt(72).

Original entry on oeis.org

1, 2, 33, 68, 1121, 2310, 38081, 78472, 1293633, 2665738, 43945441, 90556620, 1492851361, 3076259342, 50713000833, 104502261008, 1722749176961, 3550000614930, 58522759015841, 120595518646612, 1988051057361633, 4096697633369878, 67535213191279681

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).

Cf. A041126, A040063, A020829, A010524.

I:=[1, 2, 33, 68]; [n le 4 select I[n] else 34*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013

Denominator/@Convergents[Sqrt[72], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[(1 + 2 x - x^2)/(x^4 - 34 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)
a0[n_] := ((3+2*Sqrt[2])/(17+12*Sqrt[2])^n+(3-2*Sqrt[2])*(17+ 12*Sqrt[2])^n)/6 // Simplify
a1[n_] := (-1/(17+12*Sqrt[2])^n+(17+12*Sqrt[2])^n)/(12*Sqrt[2]) // FullSimplify
Flatten[MapIndexed[{a0[#],a1[#]}&,Range[20]]] (* Gerry Martens, Jul 10 2015 *)

a(n)=my(v=contfrac(sqrt(72),n),s=v[n]);forstep(k=n-1,1,-1,s=v[k]+1/s);denominator(s) \\ Charles R Greathouse IV, Jul 05 2011

G.f.: -(x^2-2*x-1) / ((x^2-6*x+1)*(x^2+6*x+1)). - Colin Barker, Nov 13 2013
a(n) = 34*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 11 2013
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = ((3+2*sqrt(2))/(17+12*sqrt(2))^n+(3-2*sqrt(2))*(17+12*sqrt(2))^n)/6.
a1(n) = (-1/(17+12*sqrt(2))^n+(17+12*sqrt(2))^n)/(12*sqrt(2)). (End)

A386461 Decimal expansion of the surface area of a biaugmented truncated cube with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 23 2025

The biaugmented truncated cube is Johnson solid J_67.

			36.241911729260269564523295159701074096328596018257...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Biaugmented truncated cube.

Cf. A010524 (volume - 9).

Cf. A010469, A010487, A010502, A377342, A385535, A386460.

First[RealDigits[18 + 8*Sqrt[2] + Sqrt[48], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J67", "SurfaceArea"], 10, 100]]

Equals 2*(9 + 4*sqrt(2) + 2*sqrt(3)) = 2*(9 + A010487 + A010469) = 18 + A377342 + A010502.

Equals the largest root of x^4 - 72*x^3 + 1592*x^2 - 10656*x - 2672.

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

A381686 Decimal expansion of the isoperimetric quotient of a truncated cube.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 04 2025

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

			0.61302821107928032110240558144714079708976169223933...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A377298 (surface area), A377299 (volume).

Cf. A000796, A002194, A010524, A156164, A381684.

First[RealDigits[49*Pi*(17 + 12*Sqrt[2])/(2*(6 + 6*Sqrt[2] + Sqrt[3])^3), 10, 100]]

Equals 36*Pi*A377299^2/(A377298^3).

Equals 49*Pi*(17 + 12*sqrt(2))/(2*(6 + 6*sqrt(2) + sqrt(3))^3) = 49*A000796*A156164/(2*(6 + A010524 + A002194)^3).

cp's OEIS Frontend

A377298 Decimal expansion of the surface area of a truncated cube with unit edge length.

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A377345 Decimal expansion of the circumradius of a truncated cuboctahedron (great rhombicuboctahedron) with unit edge length.

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A377346 Decimal expansion of the midradius of a truncated cuboctahedron (great rhombicuboctahedron) with unit edge length.

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A041126 Numerators of continued fraction convergents to sqrt(72).

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A041127 Denominators of continued fraction convergents to sqrt(72).

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A386461 Decimal expansion of the surface area of a biaugmented truncated cube with unit edges.

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A178988 Decimal expansion of volume of golden tetrahedron.

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A381686 Decimal expansion of the isoperimetric quotient of a truncated cube.

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