A239798 -id:A239798 - cp's OEIS frontend

A096748 Expansion of (1+x)^2/(1-x^2-x^4).

1, 2, 2, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 26, 34, 42, 55, 68, 89, 110, 144, 178, 233, 288, 377, 466, 610, 754, 987, 1220, 1597, 1974, 2584, 3194, 4181, 5168, 6765, 8362, 10946, 13530, 17711, 21892, 28657, 35422, 46368, 57314, 75025, 92736, 121393, 150050

Offset: 0

Paul Barry, Jul 07 2004

The ratio a(n+1) / a(n) increasingly approximates two constants connected to the golden ratio phi = (1 + sqrt(5))/2: (phi+1)/2 = 1.30901699... = A239798 and (phi-1)*2 = 1.23606797... = A134972, according to whether n is odd or even. - Davide Rotondo, Jul 31 2020

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Davide Rotondo, Perfino I Capelli Sono Tutti Contati (in Italian), see p. 11.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A000045, A079977.

Cf. A134972 and A239798 (limiting ratios for a(n+1)/a(n)).

CoefficientList[Series[(1+x)^2/(1-x^2-x^4),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,0,1},{1,2,2,2},50] (* Harvey P. Dale, Jan 29 2012 *)

a(n) = a(n-2) + a(n-4).
a(n) = 2*F((n+1)/2)*(1-(-1)^n)/2 + F((n+4)/2)*(1+(-1)^n)/2.
a(2*n) = A000045(n+2); a(2*n+1) = 2*A000045(n+1).
a(n) = Sum_{k=0..n} binomial(floor((n-k)/2), floor(k/2)). - Paul Barry, Jul 24 2004
a(n) = A079977(n) + A079977(n-2) + 2*A079977(n-1). - R. J. Mathar, Jul 15 2013

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.

A094884 Decimal expansion of phi/sqrt(2), where phi = (1+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 15 2004

An algebraic number with minimal polynomial 4*x^4 - 6*x^2 + 1. - Charles R Greathouse IV, Mar 25 2014

			1.144122805635368595200145567160604153072306753675541225...

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A001622 (phi), A002193 (sqrt(2)), A017329, A094887, A239798.

SetDefaultRealField(RealField(100)); R:= RealField(); (1+Sqrt(5) )/(2*Sqrt(2)); // G. C. Greubel, Sep 27 2018

RealDigits[GoldenRatio/Sqrt[2],10,120][[1]] (* Harvey P. Dale, Feb 11 2015 *)

sqrt(sqrt(5)+3)/2 \\ Charles R Greathouse IV, Mar 25 2014

Equals Product_{k>=0} (1 + (-1)^k/(10*k+5)). - Amiram Eldar, Nov 23 2024

Equals A094887/2 = sqrt(A239798). - Hugo Pfoertner, Nov 23 2024

A179999 Length of the n-th term in the modified Look and Say sequence A110393.

Original entry on oeis.org

1, 2, 2, 4, 6, 8, 10, 14, 18, 24, 30, 40, 50, 66, 82, 108, 134, 176, 218, 286, 354, 464, 574, 752, 930, 1218, 1506, 1972, 2438, 3192, 3946, 5166, 6386, 8360, 10334, 13528, 16722, 21890, 27058, 35420, 43782, 57312, 70842, 92734, 114626, 150048

Offset: 1

Nathaniel Johnston, Jan 13 2011

The average multiplicative growth from the n-th term to the (n+1)-st term is sqrt(phi) = 1.272..., where phi = (1+sqrt(5))/2 is the golden ratio, see A139339.

			The 6th term in A110393 is 21112211, so a(6) = 8.

Colin Barker, Table of n, a(n) for n = 1..1000
N. Johnston, Further Variants of the “Look-and-Say” Sequence
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1).

Cf. A005341, A049194, A098596, A110393.

Cf. A001622, A134972, A139339, A239798.

CoefficientList[Series[((1+x) (-1-x+x^2) (1-x+x^2))/((1-x) (-1+x^2+x^4)),{x,0,99}],x] (* Peter J. C. Moses, Jun 23 2013 *)

Vec(x*(1 + x)*(1 + x - x^2)*(1 - x + x^2) / ((1 - x)*(1 - x^2 - x^4)) + O(x^50)) \\ Colin Barker, Aug 10 2019

a(n) = length(A110393(n)).
From Colin Barker, Aug 10 2019: (Start)
G.f.: x*(1 + x)*(1 + x - x^2)*(1 - x + x^2) / ((1 - x)*(1 - x^2 - x^4)).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n>6. (End)
From A.H.M. Smeets, Aug 10 2019 (Start)
Limit_{n->oo} a(n+1)/a(n) = (1+phi)/2 = (3+sqrt(5))/4 = A239798 for odd n.
Limit_{n->oo} a(n+1)/a(n) = 2/phi = 4/(1+sqrt(5)) = A134972 for even n.
Limit_{n->oo} a(n+2)/a(n) = (1+phi)/phi = phi = A001622. (End)
For odd n > 1, a(n) = 4*Fibonacci((n + 1)/2) - 2. For even n, a(n) = 2*Fibonacci(n/2 + 2) - 2. - Ehren Metcalfe, Aug 10 2019

A341906 Decimal expansion of the moment of inertia of a solid regular dodecahedron with a unit mass and a unit edge length.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 04 2021

The moments of inertia of the five Platonic solids were apparently first calculated by the Canadian physicist John Satterly (1879-1963) in 1957.
The moment of inertia of a solid regular dodecahedron with a uniform mass density distribution, mass M, and edge length L is I = c*M*L^2, where c is this constant.
The corresponding values of c for the other Platonic solids are:
Tetrahedron: 1/20 (= A020761/10).
Octahedron: 1/10 (= A000007).
Cube: 1/6 (= A020793).
Icosahedron: (3 + sqrt(5))/20 (= A104457/10).

			0.60735550374163932719985924360173257727394705341616...

P. K. Aravind, Gravitational collapse and moment of inertia of regular polyhedral configurations, American Journal of Physics, Vol. 59, No. 7 (1991), pp. 647-652.
Frédéric Perrier, Moments of inertia of Archimedean solids, 2015.
John Satterly, Moments of Inertia about Selected Axes of Regular Polygons, Right Pyramids on Regular Polygonal Bases, and of the Platonic and Some Archimedian Polyhedra, American Journal of Physics, Vol. 25, No. 7 (1957), pp. 489-490.
John Satterly, The Moments of Inertia of Some Polyhedra, The Mathematical Gazette, Vol. 42, No. 339 (1958), pp. 11-13.
Wikipedia, List of moments of inertia.
Wikipedia, Moment of inertia.
Wikipedia, Regular dodecahedron.
Index entries for sequences related to moment of inertia.

Cf. A000007, A001622, A020761, A020793, A104457.

Other constants related to the regular dodecahedron: A102769, A131595, A179296, A232810, A237603, A239798.

RealDigits[(95 + 39*Sqrt[5])/300, 10, 100][[1]]

Equals (95 + 39*sqrt(5))/300.

Equals (28 + 39*phi)/150, where phi is the golden ratio (A001622).

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A096748 Expansion of (1+x)^2/(1-x^2-x^4).

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

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A094884 Decimal expansion of phi/sqrt(2), where phi = (1+sqrt(5))/2.

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A179999 Length of the n-th term in the modified Look and Say sequence A110393.

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A341906 Decimal expansion of the moment of inertia of a solid regular dodecahedron with a unit mass and a unit edge length.

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