A118273 -id:A118273 - cp's OEIS frontend

A122553 a(0)=1, a(n)=3 for n > 0.

1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Philippe Deléham, Sep 20 2006

Continued fraction for (sqrt(13) - 1)/2 = A223139.
Decimal expansion of 4/30. - Alonso del Arte, Aug 16 2012
4/3 is the volume of the regular octahedron inscribed in the unit-radius sphere. - Amiram Eldar, Jun 02 2023

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Springer, 2013, pp. 95-96, 224.

Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A118273 (cube), A339259 (regular icosahedron), A363437 (regular tetrahedron), A363438 (regular dodecahedron).

Cf. A223139.

RealDigits[4/3, 10, 105][[1]] (* Alonso del Arte, Aug 16 2012 *)
PadRight[{1},120,3] (* Harvey P. Dale, Jul 21 2023 *)

a(n)=(n>=0)+2*(n>0) \\ Jaume Oliver Lafont, Mar 26 2009

a(n) = 3 - 2*0^n.
G.f.: (1 + 2*x)/(1 - x).
Sum_{n >= 0} a(n)*10^(-n) = 4/3.
From Amiram Eldar, Jun 05 2021: (Start)
4/3 = Product_{k>=1} (1 + 1/2^(2^k)).
4/3 = Sum_{k>=0} binomial(2*k,k)/((k+2)*4^k). (End)
Sum_{k>0} 3*k/4^k = 4/3 [Nicole Oresme]. - Stefano Spezia, Jun 27 2024
K_{n>=3} n/(n-2) = 4/3 (see Clawson at p. 224). - Stefano Spezia, Jul 01 2024
E.g.f.: 3*exp(x) - 2. - Elmo R. Oliveira, Aug 05 2024

A339259 Decimal expansion of the volume of the regular icosahedron inscribed in the unit sphere.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Nov 29 2020

			2.536150710120409525643838222345019049081863024335333926526148385147...

Index entries for algebraic numbers, degree 4.

Cf. A001622, A019881, A102208.

Cf. A118273 (cube), A122553 (regular octahedron), A363437 (regular tetrahedron), A363438 (regular dodecahedron).

RealDigits[4 * Sqrt[GoldenRatio + 2]/3, 10, 120][[1]] (* Amiram Eldar, Jun 02 2023 *)

PARI
```
4/3*sqrt(2+(1+sqrt(5))/2)
```

Equals 4*sqrt(2 + phi)/3 where phi = A001622.

Equals A102208 / A019881 ^ 3. - Amiram Eldar, Jun 02 2023

A327494 a(n) = numerator(r(n)) where r(n) = Sum_{j=0..n} j!/(2^j*floor(j/2)!)^2.

Original entry on oeis.org

1, 5, 11, 47, 191, 779, 1563, 6287, 50331, 201639, 403341, 1614057, 6456459, 25828839, 51658107, 206638863, 3306228243, 13225022367, 26450056889, 105800458501, 423201880193, 1692808490741, 3385617069661, 13542470306761, 108339763130127, 433359069421483

Offset: 0

Peter Luschny, Sep 27 2019

			r(n) = 1, 5/4, 11/8, 47/32, 191/128, 779/512, 1563/1024, 6287/4096, 50331/32768, 201639/131072, ...

Denominators are in A327493.

Cf. A327491, A327492, A327493, A327495, A118273, A163590.

A327494(n) = sum(<<(A163590(k), A327492(n) - A327492(k)) for k in 0:n) # Peter Luschny, Oct 03 2019

A327494 := n -> numer(add(j!/(2^j*iquo(j,2)!)^2, j=0..n)):
seq(A327494(n), n=0..25);

a(n)={ numerator(sum(j=0, n, j!/(2^j*(j\2)!)^2)) } \\ Andrew Howroyd, Sep 28 2019

Lim_{n -> oo} r(n) = (4/3)^(3/2) = A118273.

A054759 Number of Eulerian orientations of the n X n square lattice (with wrap-around), i.e., number of arrow configurations on n X n grid that satisfy the square ice rule.

Original entry on oeis.org

4, 18, 148, 2970, 143224, 16448400, 4484823396, 2901094068042, 4448410550095612, 16178049740086515288, 139402641051212392498528, 2849295959501939989625992464, 137950545200232788276834783781648, 15844635835975276495290739119895808472

Offset: 1

Steven Finch, Apr 25 2000

nonn

The n X n square lattice with wrap around is also called the torus grid graph. - Andrew Howroyd, Jan 11 2018

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 412-416.
Computed by Jennifer Henry in Dec. 1998.

E. H. Lieb, Residual entropy of square ice, Phys. Rev. 162 (1967) 162-172.
Steven R. Finch, Lieb's Square Ice Constant [Broken link]
Steven R. Finch, Lieb's Square Ice Constant [From the Wayback machine]
Eric Weisstein's World of Mathematics, Torus Grid Graph

Cf. A118273, A358177. Main diagonal of A298119.

Elliot Lieb proved that lim_{n->oo} a(n)^(1/n^2) = (4/3)^(3/2). See A118273.

a(14) from Brendan McKay, Apr 18 2024

A363437 Decimal expansion of the volume of the regular tetrahedron inscribed in the unit-radius sphere.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 02 2023

			0.51320023927966734623035447155729551613120155668455...

Eric Weisstein's World of Mathematics, Regular Tetrahedron.
Wikipedia, Tetrahedron: Regular tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A118273 (cube), A122553 (regular octahedron), A339259 (regular icosahedron), A363438 (regular dodecahedron).

Other constants related to the regular tetrahedron: A020781, A020829, A137914, A156546, A187110, A210974, A232812, A236555.

Mathematica
```
RealDigits[8/(9*Sqrt[3]), 10, 120][[1]]
```

8/9/sqrt(3) \\ Charles R Greathouse IV, Feb 07 2025

Equals 8/(9*sqrt(3)).
Equals A118273 / 3.
Equals A020829 / A187110 ^ 3.

A363438 Decimal expansion of the volume of the regular dodecahedron inscribed in the unit-radius sphere.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 02 2023

			2.78516386312262296729255491273594698789932177207633...

Eric Weisstein's World of Mathematics, Regular Dodecahedron.
Wikipedia, Regular dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A118273 (cube), A122553 (regular octahedron), A339259 (regular icosahedron), A363437 (regular tetrahedron).
Cf. A001622.
Other constants related to the regular dodecahedron: A102769, A131595, A179296, A232810, A237603, A239798, A341906.

RealDigits[(2*(5 + Sqrt[5]))/(3*Sqrt[3]), 10, 120][[1]]

2*sqrt(5+sqrt(5))/sqrt(27) \\ Charles R Greathouse IV, Feb 07 2025

Equals 2*sqrt(5+sqrt(5))/(3*sqrt(3)).
Equals 4*(phi+2)/(3*sqrt(3)), where phi is the golden ratio (A001622).
Equals A102769 / A179296 ^ 3.

A256888 Terms of the continued fraction expansion of 1 + sqrt(64 / 37).

Original entry on oeis.org

2, 3, 5, 1, 3, 1, 3, 1, 5, 3, 2, 3, 5, 1, 3, 1, 3, 1, 5, 3, 2, 3, 5, 1, 3, 1, 3, 1, 5, 3, 2, 3, 5, 1, 3, 1, 3, 1, 5, 3, 2, 3, 5, 1, 3, 1, 3, 1, 5, 3, 2, 3, 5, 1, 3, 1, 3, 1, 5, 3, 2, 3, 5, 1, 3, 1, 3, 1, 5, 3, 2, 3, 5, 1, 3, 1, 3, 1, 5, 3, 2, 3, 5, 1, 3, 1, 3, 1, 5, 3

Offset: 1

Derek V. Schmalenberger, Apr 12 2015

The number sqrt(64/37) is of interest because "Lieb's square ice constant" is sqrt(64/27) and 27+37 = 64 and 27*37 = 999. Adding 1 to sqrt(64/37) creates a continued fraction with a cycle length of 10.

Wikipedia, Lieb's square ice constant

Cf. A118273.

ContinuedFraction[1 + Sqrt[64/37], 120] (* Michael De Vlieger, Apr 24 2015 *)

cp's OEIS Frontend

A122553 a(0)=1, a(n)=3 for n > 0.

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A339259 Decimal expansion of the volume of the regular icosahedron inscribed in the unit sphere.

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A327494 a(n) = numerator(r(n)) where r(n) = Sum_{j=0..n} j!/(2^j*floor(j/2)!)^2.

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A054759 Number of Eulerian orientations of the n X n square lattice (with wrap-around), i.e., number of arrow configurations on n X n grid that satisfy the square ice rule.

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A363437 Decimal expansion of the volume of the regular tetrahedron inscribed in the unit-radius sphere.

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A363438 Decimal expansion of the volume of the regular dodecahedron inscribed in the unit-radius sphere.

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A256888 Terms of the continued fraction expansion of 1 + sqrt(64 / 37).

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