A020797 -id:A020797 - cp's OEIS frontend

A295333 Numerators of continued fraction convergents to sqrt(10)/2 = sqrt(5/2) = A020797 + 1.

1, 2, 3, 8, 11, 19, 49, 68, 117, 302, 419, 721, 1861, 2582, 4443, 11468, 15911, 27379, 70669, 98048, 168717, 435482, 604199, 1039681, 2683561, 3723242, 6406803, 16536848, 22943651, 39480499, 101904649, 141385148, 243289797, 627964742, 871254539, 1499219281, 3869693101, 5368912382, 9238605483

Offset: 0

Wolfdieter Lang, Nov 21 2017

The denominators are given in A295334.

The regular continued fraction expansion of sqrt(10)/2 is [1, repeat(1, 1, 2)].

			The convergents a(n)/A295334(n) begin: 1, 2, 3/2, 8/5, 11/7, 19/12, 49/31, 68/43, 117/74, 302/191, 419/265, 721/456, 1861/1177, 2582/1633, 4443/2810, 11468/7253, 15911/10063, 27379/17316, 70669/44695, 98048/62011, ...

Robert Israel, Table of n, a(n) for n = 0..3794
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,1).

Cf. A020797, A295334.

numtheory:-cfrac(sqrt(5/2),100,'con'):
map(numer,con[1..-2]); # Robert Israel, Nov 22 2017

Numerator[Convergents[Sqrt[5/2], 50]] (* Vaclav Kotesovec, Nov 22 2017 *)
LinearRecurrence[{0,0,6,0,0,1},{1,2,3,8,11,19},40] (* Harvey P. Dale, Apr 08 2019 *)

G.f.: G(x) = (1 + 2*x + 3*x^2 + 2*x^3 - x^4 + x^5)/(1 - 6*x^3 - x^6). From the recurrence a(n) = b(n)*a(n-1) + a(n-2), with the trisection b(3*(k+1)) = 2, b(3*k+1) = 1 = b(3*k+2), k >= 0, b(0) = 1, and the input a(0) = 1 = a(-1). With G_j(x) = Sum_{k>=0} a(3*k+j)*x^k, for j = 0,1,2, one finds (omitting here the G_j arguments) G_0 = 1 + 2*x*G_2 + x*G_1, G_1 = G_0 + 1 + x*G_2, G_2 = G_1 + G_0. This can be solved and leads to the given formula for G(x) = Sum_{j=0..2} x^j*G_j(x^3).

a(n) = 6*a(n-3) + a(n-6), for n >= 6, with inputs a(0)..a(5).

A295334 Denominators of continued fraction convergents to sqrt(10)/2 = sqrt(5/2) = A020797 + 1.

Original entry on oeis.org

1, 1, 2, 5, 7, 12, 31, 43, 74, 191, 265, 456, 1177, 1633, 2810, 7253, 10063, 17316, 44695, 62011, 106706, 275423, 382129, 657552, 1697233, 2354785, 4052018, 10458821, 14510839, 24969660, 64450159, 89419819, 153869978, 397159775, 551029753, 948189528, 2447408809, 3395598337, 5843007146

Offset: 0

Wolfdieter Lang, Nov 21 2017

The numerators are given in A295333. There details are given.

			For the first convergents see A295333.

Robert Israel, Table of n, a(n) for n = 0..3795
Index entries for linear recurrences with constant coefficients, signature (0, 0, 6, 0, 0, 1).

Cf. A020797, A295333.

numtheory:-cfrac(sqrt(5/2),100,'con'):
map(denom,con[1..-2]); # Robert Israel, Nov 22 2017

Denominator[Convergents[Sqrt[5/2], 50]] (* Wesley Ivan Hurt, Nov 21 2017 *)

G.f.: G(x) = (1 + x + 2*x^2 - x^3 + x^4)/(1 - 6*x^3 - x^6), For the derivation see A295333, but here the input of the recurrence is a(0) = 1, a(-1) = 0 (a(-2) = a(0) = 1). This leads here to G_0 = 1+ 2*x*G_2 + x*G_1, G_1 = G_0 + x*G_2, G_2 = G_1 + G_0 and the solution gives G(x).

a(n) = 6*a(n-3) + a(n-6), n >= 6, with inputs a(0)..a(5).

A377342 Decimal expansion of the volume of a truncated octahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			11.3137084989847603904135097936775846285573750030...

Eric Weisstein's World of Mathematics, Truncated Octahedron.
Wikipedia, Truncated octahedron.

Cf. A377341 (surface area), A020797 (circumradius/10), A152623 (midradius).
Cf. A131594 (analogous for a regular octahedron).
Cf. A002193, A010466, A010487.

First[RealDigits[8*Sqrt[2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedOctahedron", "Volume"], 10, 100]]

Equals 8*sqrt(2) = 8*A002193 = 4*A010466 = 2*A010487.

A041066 Numerators of continued fraction convergents to sqrt(40).

Original entry on oeis.org

6, 19, 234, 721, 8886, 27379, 337434, 1039681, 12813606, 39480499, 486579594, 1499219281, 18477210966, 56930852179, 701647437114, 2161873163521, 26644125399366, 82094249361619, 1011775117738794

Offset: 0

N. J. A. Sloane

With a(-1) = 1, a(n-1) gives, for n >= 0, the denominator of the convergents to 1/sqrt(40) = 1/(2*sqrt(10)). - Wolfdieter Lang, Nov 21 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).

Cf. A041067 (denominators), A010494, A020797 (1/sqrt(40)).

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[40],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)
Numerator[Convergents[Sqrt[40], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
LinearRecurrence[{0,38,0,-1},{6,19,234,721},20] (* Harvey P. Dale, Jul 20 2024 *)

G.f.: -(x+2)*(x^2-8*x-3) / ((x^2-6*x-1)*(x^2+6*x-1)). - Colin Barker, Nov 04 2013

A041067 Denominators of continued fraction convergents to sqrt(40).

Original entry on oeis.org

1, 3, 37, 114, 1405, 4329, 53353, 164388, 2026009, 6242415, 76934989, 237047382, 2921503573, 9001558101, 110940200785, 341822160456, 4212806126257, 12980240539227, 159975692596981, 492907318330170, 6074863512559021, 18717497856007233, 230684837784645817

Offset: 0

N. J. A. Sloane

With a(-1) = 0, a(n-1) gives, for n >= 0, the numerator of the convergents to 1/sqrt(40) = 1/(2*sqrt(10)) = A020797. - Wolfdieter Lang, Nov 21 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).

Cf. A010494, A041066 (numerators).

I:=[1, 3, 37, 114]; [n le 4 select I[n] else 38*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[40],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011 *)
Denominator[Convergents[Sqrt[40],30]] (* Harvey P. Dale, Sep 12 2013 *)
CoefficientList[Series[-(x^2 - 3 x - 1)/((x^2 - 6 x - 1)(x^2 + 6 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2013 *)

G.f.: -(x^2-3*x-1) / ((x^2-6*x-1)*(x^2+6*x-1)). - Colin Barker, Nov 12 2013

a(n) = 38*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013

More terms from Colin Barker, Nov 12 2013

A377341 Decimal expansion of the surface area of a truncated octahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			26.78460969082652752232935609807046840331366304572...

Eric Weisstein's World of Mathematics, Truncated Octahedron.
Wikipedia, Truncated octahedron.

Cf. A377342 (volume), A020797 (circumradius/10), A152623 (midradius).
Cf. A010469 (analogous for a regular octahedron).
Cf. A002194.

First[RealDigits[6 + 12*Sqrt[3], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedOctahedron", "SurfaceArea"], 10, 100]]

Equals 6 + 12*sqrt(3) = 6 + 12*A002194.

A041468 Numerators of continued fraction convergents to sqrt(250).

Original entry on oeis.org

15, 16, 79, 253, 838, 3605, 4443, 136895, 141338, 702247, 2248079, 7446484, 32034015, 39480499, 1216448985, 1255929484, 6240166921, 19976430247, 66169457662, 284654260895, 350823718557, 10809365817605, 11160189536162, 55450123962253, 177510561422921

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,8886,0,0,0,0,0,0,1).

Cf. A020797, A041469.

Numerator[Convergents[Sqrt[250], 30]] (* Vincenzo Librandi, Nov 03 2013 *)

G.f.: -(x^13 -15*x^12 +16*x^11 -79*x^10 +253*x^9 -838*x^8 +3605*x^7 +4443*x^6 +3605*x^5 +838*x^4 +253*x^3 +79*x^2 +16*x +15)/(x^14 +8886*x^7 -1). - Colin Barker, Nov 08 2013

More terms from Colin Barker, Nov 08 2013

A188924 Decimal expansion of sqrt(4+sqrt(15)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a sqrt(6)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A sqrt(6)-extension rectangle matches the continued fraction [2,1,4,6,1,1,2,25,3,...] for the shape L/W=sqrt(4+sqrt(15)). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the sqrt(6)-extension rectangle, 2 squares are removed first, then 1 square, then 4 squares, then 6 squares,..., so that the original rectangle of shape sqrt(4+sqrt(15)) is partitioned into an infinite collection of squares.

			2.8058837014757787150980888095693049628427513...

Cf. A020797, A115754, A188925.

r = 6^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

sqrt(3/2) + sqrt(5/2) \\ Hugo Pfoertner, Feb 20 2024

Equals A115754 + 10*A020797. - Hugo Pfoertner, Feb 20 2024

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A295333 Numerators of continued fraction convergents to sqrt(10)/2 = sqrt(5/2) = A020797 + 1.

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A295334 Denominators of continued fraction convergents to sqrt(10)/2 = sqrt(5/2) = A020797 + 1.

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A377342 Decimal expansion of the volume of a truncated octahedron with unit edge length.

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A041066 Numerators of continued fraction convergents to sqrt(40).

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A041067 Denominators of continued fraction convergents to sqrt(40).

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A377341 Decimal expansion of the surface area of a truncated octahedron with unit edge length.

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A041468 Numerators of continued fraction convergents to sqrt(250).

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A188924 Decimal expansion of sqrt(4+sqrt(15)).

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