A197723 -id:A197723 - cp's OEIS frontend

A299798 Ground state degeneracy of a periodic chain for angle theta = 3*Pi/2.

Original entry on oeis.org

26, 72, 198, 570, 1641, 4725, 13605, 39174

Offset: 3

Michael De Vlieger, Feb 19 2018

Yun-Tak Oh, Hosho Katsura, Hyun-Yong Lee, Jung Hoon Han, Proposal of a spin-one chain model with competing dimer and trimer interactions, arXiv:1709.01344 [cond-mat.str-el], 2017.

Cf. A005248, A197723, A299797.

A332331 Decimal expansion of the next-to-least positive zero of the 12th Maclaurin polynomial of cos x.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 11 2020

The Maclaurin polynomial p(2n,x) of cos x is 1 - x^2/2! + x^4/4! + ... + (-1)^n x^(2n)/(2n)!.

Let z(n) be the next-to-least positive zero of p(2n,x) if there is such a zero. The limit of z(n) is 3 Pi/2 = 4.7123889..., as in A197723.

			Next-to-least positive zero = 4.6865176637957574465700489837907750...

Cf. A197723, A332329, A332330.

z = 150; p[n_, x_] := Normal[Series[Cos[x], {x, 0, n}]]
t = x /. NSolve[p[12, x] == 0, x, z][[8]]
u = RealDigits[t][[1]]
Plot[Evaluate[p[12, x]], {x, -1, 5}]

A359104 Decimal expansion of the area enclosed by Sylvester's Bicorn curve.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Dec 18 2022

The Cartesian equation of Sylvester's Bicorn curve is y^2*(m^2-x^2) = (x^2+2*m*y-m^2)^2, here with parameter m=1. The area is proportional to the square m^2 of parameter m.

Corresponding arc length is given by A228764.

			0.746455945439346463341461672758965758770535375107896820343...

M. Protat, Des Olympiades à l'Agrégation, Encadrement du bicorne, Problème 66, pp. 142-145, Ellipses, Paris 1997.

Robert Ferréol, Bicorn, Mathcurve.
Eric Weisstein's World of Mathematics, Bicorn.
Wikipedia, Bicorn.
Index to sequences related to curves.

Cf. A228764 (length).

Other area of curves: A019692 (deltoid), A197723 (cardioid), A122952 (nephroid), A180434 (Newton strophoid).

Maple
```
evalf((16*sqrt(3) - 27)*Pi/3, 100);
```

RealDigits[(16*Sqrt[3] - 27)*Pi/3, 10, 120][[1]] (* Amiram Eldar, Dec 18 2022 *)

Equals (16*sqrt(3) - 27)*Pi/3.

cp's OEIS Frontend

A299798 Ground state degeneracy of a periodic chain for angle theta = 3*Pi/2.

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A332331 Decimal expansion of the next-to-least positive zero of the 12th Maclaurin polynomial of cos x.

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A359104 Decimal expansion of the area enclosed by Sylvester's Bicorn curve.

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Mathematica

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