A176394 -id:A176394 - cp's OEIS frontend

A343964 Decimal expansion of 18 + 2*sqrt(3).

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

Offset: 1

Wesley Ivan Hurt, May 05 2021

Surface area of a rhombicuboctahedron with unit edge length.

Essentially the same sequence of digits as A176394 and A010469. - R. J. Mathar, May 07 2021

			21.464101615137754587054892683011744733885...

Wikipedia, Rhombicuboctahedron

Cf. A343965 (rhombicuboctahedron volume).

SetDefaultRealField(RealField(100)); 18+2*Sqrt(3);

RealDigits[N[18 + 2*Sqrt[3], 100]][[1]] (* Wesley Ivan Hurt, Nov 12 2022 *)

A343199 Decimal expansion of 6+2*sqrt(3).

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 07 2021

Decimal expansion of the surface area of a cuboctahedron with unit edge length.

Essentially the same sequence of digits as A176394 and A010469. - R. J. Mathar, Jun 10 2021

			9.4641016151377545870548926830117447338856...

Wikipedia, Cuboctahedron

Cf. A020775 (cuboctahedron volume).

SetDefaultRealField(RealField(200)); 6+2*Sqrt(3);

RealDigits[N[6 + 2*Sqrt[3], 100]][[1]] (* Wesley Ivan Hurt, Nov 12 2022 *)

A286760 Total number of nodes summed over all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps U=(1,1), D=(1,-1), H=(1,0) and S=(0,1).

Original entry on oeis.org

1, 2, 10, 42, 214, 1098, 5978, 33190, 189078, 1093490, 6414714, 38027030, 227489950, 1370980490, 8314674202, 50696630838, 310541818382, 1909850054666, 11786947172234, 72969941803662, 452976340653030, 2818815920369754, 17579546535174946, 109850944544149134

Offset: 0

Alois P. Heinz, May 14 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A176394, A225041.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1$2],
      (p-> p+[0, p[1]])(b(x-1, y)+b(x, y-1)+b(x-1, y-1)+b(x-1, y+1))))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);

b[x_, y_] := b[x, y] = If[y<0 || y>x, 0, If[x == 0, {1, 1}, Function[
   p, p+{0, p[[1]]}][b[x-1, y] + b[x, y-1] + b[x-1, y-1] + b[x-1, y+1]]]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 28 2022, after Alois P. Heinz *)

a(n) ~ c * (3 + 2*sqrt(3))^n / sqrt(n), where c = 0.0889843039487036085233000284915570190371055498671732340656... - Vaclav Kotesovec, Sep 11 2021

A277390 Decimal expansion of 5-2sqrt(5)+sqrt(25-10sqrt(5))-sqrt(5-2*sqrt(5)).

Original entry on oeis.org

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

Offset: 1

Martin Renner, Oct 12 2016

Largest radius of five circles tangent to a circle of radius 1.

A quartic integer with minimal polynomial x^4 - 20x^3 + 10x^2 + 20x + 5. - Charles R Greathouse IV, Oct 12 2016

			1.42591999815959135206...

Eric W. Weisstein: Steiner chain. From MathWorld. A Wolfram Web Resource.
Index entries for algebraic numbers, degree 4

Cf. A176394 (three circles), A014176 (four circles).

RealDigits[5-2*Sqrt[5]+Sqrt[25-10*Sqrt[5]]-Sqrt[5-2*Sqrt[5]],10,120][[1]] (* Harvey P. Dale, May 20 2021 *)

s=sqrt(5); t=5-2*s; sqrt(25-10*s)+t-sqrt(t) \\ Charles R Greathouse IV, Oct 12 2016

tan(Pi/5)*(tan(Pi/5)+sqrt(1+tan(Pi/5)^2)).

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A343964 Decimal expansion of 18 + 2*sqrt(3).

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A343199 Decimal expansion of 6+2*sqrt(3).

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A286760 Total number of nodes summed over all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps U=(1,1), D=(1,-1), H=(1,0) and S=(0,1).

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A277390 Decimal expansion of 5-2*sqrt(5)+sqrt(25-10*sqrt(5))-sqrt(5-2*sqrt(5)).

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A277390 Decimal expansion of 5-2sqrt(5)+sqrt(25-10sqrt(5))-sqrt(5-2*sqrt(5)).