A204188 -id:A204188 - cp's OEIS frontend

A386464 Decimal expansion of the volume of an augmented truncated dodecahedron with unit edges.

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

Offset: 2

Paolo Xausa, Jul 25 2025

The augmented truncated dodecahedron is Johnson solid J_68.

			87.3637098777040746856191001251416771010058551...

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

Cf. A386465 (surface area).

Cf. A179590, A204188, A377695, A386411, A386459, A386466, A386544.

First[RealDigits[505/12 + 81/4*Sqrt[5], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J68", "Volume"], 10, 100]]

Equals 505/12 + 81*sqrt(5)/4 = 505/12 + 81*A204188.
Equals A377695 + A179590.
Equals the largest root of 36*x^2 - 3030*x - 10055.

A375804 a(n) = Lucas(n-1) * Lucas(n+1) * Fibonacci(2n-1) Fibonacci(2*n+1).

Original entry on oeis.org

12, 40, 1365, 19448, 381276, 6615103, 120241980, 2147070680, 38600066517, 692153278024, 12423591148332, 222908960952575, 4000098954110700, 71777766990248968, 1288007282149222101, 23112301389881302808, 414733773612913239420, 7442093184423393874495, 133542960264663589170972

Offset: 1

Amiram Eldar, Aug 29 2024

Amiram Eldar, Table of n, a(n) for n = 1..798
Hideyuki Ohtskua, proposer, Problem H-944, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 62, No. 3 (2024), p. 266.
Index entries for linear recurrences with constant coefficients, signature (13,104,-260,-260,104,13,-1).

Cf. A000032, A000045, A064170, A204188, A292696, A375803.

a[n_] := LucasL[n-1] * LucasL[n+1] * Fibonacci[2*n-1] * Fibonacci[2*n+1]; Array[a, 20]

lucas(n) = fibonacci(n-1) + fibonacci(n+1);
a(n) = lucas(n-1) * lucas(n+1) * fibonacci(2*n-1) * fibonacci(2*n+1);

a(n) = A292696(n) * A064170(n+2).
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(5) - 2)/ 4 = A204188 - 1/2 (Ohtskua, 2024).
G.f.: -x^2*(-20+65*x+195*x^2-84*x^3-13*x^4+x^5)/ ( (1+x) *(x^2-3*x+1) *(x^2+7*x+1) *(x^2-18*x+1) ). - R. J. Mathar, Aug 30 2024

A356869 Decimal expansion of 4 / sqrt(5).

Original entry on oeis.org

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

Offset: 1

Michal Paulovic, Sep 01 2022

The area of a golden rectangle inscribed in a unit circle.
The width and height of the rectangle are:
  W = sqrt(2 - 2 / sqrt(5)) = A179290.
  H = sqrt(2 + 2 / sqrt(5)) = A121570.

			1.7888543819998317...

Cf. A121570, A179290, A204188.

cell2mat(struct2cell(struct(vpa(4 / sqrt(5), 105)))); ans(1:98)

parse(substring(convert(evalf(4 / sqrt(5), 105), string), 1..98));

RealDigits[4 / Sqrt[5], 10, 105][[1]][[Range[1, 97]]]

Equals [1; 1, 3, 1, 2] (periodic continued fraction expansion). - Peter Luschny, Sep 02 2022

Equals 1/A204188. - Alois P. Heinz, Sep 02 2022

A357715 Decimal expansion of sqrt(16 + 32 / sqrt(5)).

Original entry on oeis.org

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

Offset: 1

Michal Paulovic, Oct 10 2022

The perimeter of a golden rectangle inscribed in a unit circle.
The width and height of the rectangle are:
  W = sqrt(2 - 2 / sqrt(5)) = A179290.
  H = sqrt(2 + 2 / sqrt(5)) = A121570.

			5.5055276818846941...

Cf. A019934, A019952, A019970, A121570, A179290, A204188, A356869.

Maple
```
sqrt(16 + 32 / sqrt(5));
```
Mathematica
```
Sqrt[16 + 32/Sqrt[5]]
```
PARI
```
sqrt(16 + 32 / sqrt(5))
```

Equals (4 / sqrt(5)) * sqrt(5 + 2 * sqrt(5)) = A356869 * A019970.
Equals sqrt(5 + 2 * sqrt(5)) / (sqrt(5) / 4) = A019970 / A204188.
Equals 4 * sqrt(1 + 2 / sqrt(5)) = 4 * A019952.
Equals 4 / sqrt(5 - 2 * sqrt(5)) = 4 / A019934.

cp's OEIS Frontend

A386464 Decimal expansion of the volume of an augmented truncated dodecahedron with unit edges.

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Mathematica

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A375804 a(n) = Lucas(n-1) * Lucas(n+1) * Fibonacci(2*n-1) * Fibonacci(2*n+1).

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A356869 Decimal expansion of 4 / sqrt(5).

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MATLAB

Maple

Mathematica

Formula

A357715 Decimal expansion of sqrt(16 + 32 / sqrt(5)).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A375804 a(n) = Lucas(n-1) * Lucas(n+1) * Fibonacci(2n-1) Fibonacci(2*n+1).