A010499 -id:A010499 - cp's OEIS frontend

A248271 Egyptian fraction representation of sqrt(45) (A010499) using a greedy function.

6, 2, 5, 122, 138674, 32476589259, 7827697016386517458238, 674742854143668103289252692160450020023615629, 480580099090725670530151893237450499682750267119621001128141465878491826900413350973083878

Offset: 0

Robert G. Wilson v, Oct 04 2014

nonn

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.

Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 45]]

A104161 G.f.: x(1 - x + x^2)/((1-x)^2 (1 - x - x^2)).

Original entry on oeis.org

0, 1, 2, 5, 10, 19, 34, 59, 100, 167, 276, 453, 740, 1205, 1958, 3177, 5150, 8343, 13510, 21871, 35400, 57291, 92712, 150025, 242760, 392809, 635594, 1028429, 1664050, 2692507, 4356586, 7049123

Offset: 0

Creighton Dement, Mar 10 2005

A floretion-generated sequence.
Floretion Algebra Multiplication Program, FAMP Code: 1vesrokseq[ (- .25'i - .25i' - .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' - .25e)('i + i' + 'ji' + 'ki' + e) ] RokType: Y[sqa.Findk()] = Y[sqa.Findk()] + p.
Partial sums of Leonardo numbers A001595. - Jonathan Vos Post, Jan 01 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A006355, A001595, A117501, A145912.

List([0..40], n-> 2*Fibonacci(n+2) -(n+2)); # G. C. Greubel, Jul 09 2019

[2*Fibonacci(n+2) -(n+2): n in [0..40]]; // G. C. Greubel, Jul 09 2019

a=0;b=1;Table[c=b+a+n; a=b; b=c, {n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
CoefficientList[Series[x*(1-x+x^2)/((1-x)^2*(1-x-x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{3,-2,-1,1},{0,1,2,5},40] (* Harvey P. Dale, Sep 06 2012 *)

my(x='x+O('x^40)); concat(0, Vec(x*(1-x+x^2)/((1-x)^2*(1-x-x^2)))) \\ G. C. Greubel, Sep 26 2017

[2*fibonacci(n+2) -(n+2) for n in (0..40)] # G. C. Greubel, Jul 09 2019

Superseeker results (incomplete): a(2) - 2a(n+1) + a(n) = A006355(n+1) (Number of binary vectors of length n containing no singletons); a(n+1) - a(n) = A001595(n) (2-ranks of difference sets constructed from Segre hyperovals); a(n) + n + 1 = A001595(n+1).
A107909(a(n)) = A000975(n). - Reinhard Zumkeller, May 28 2005
From Ross La Haye, Aug 03 2005: (Start)
a(n) = 2*(Fibonacci(n+2) - 1) - n.
a(n) = Sum_{k=0..n} A101220(n-k, 0, k). (End)
From Gary W. Adamson, Apr 02 2006: (Start)
a(n) = a(n-1) + a(n-2) + n-1.
a(n) = row sums of A117501, starting (1, 2, 5, 10, ...). (End)
a(n) = Sum_{k=0..n} A109754(n-k,k). - Ross La Haye, Apr 12 2006
a(n) = (Sum_{k=0..n} (n-k)*Fibonacci(k-1) + Fibonacci(k)) - n. - Ross La Haye, May 31 2006
From R. J. Mathar, Apr 18 2008: (Start)
a(n) = -2 - n + (-A094214)^n*(1-A010499/5) + (1+A010499/5)/A094214^n.
a(n) = A006355(n+3) - n - 2. (End)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5. - Harvey P. Dale, Sep 06 2012

A176015 Decimal expansion of (5 + 3*sqrt(5))/10.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 06 2010

Continued fraction expansion of (5 + 3*sqrt(5))/10 is A010686.

The horizontal distance between the accumulation point and the outermost point of a golden spiral inscribed inside a golden rectangle with dimensions phi and 1 along the x and y axes, respectively (the vertical distance is A244847). - Amiram Eldar, May 18 2021

			(5 + 3*sqrt(5))/10 = 1.17082039324993690892...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.2 The Golden Mean, phi, p. 7.

Muniru A Asiru, Table of n, a(n) for n = 1..1000 [a(1000) corrected by _Georg Fischer_, Apr 02 2020]
Index entries for algebraic numbers, degree 2.

Cf. A000032, A000045, A001622, A002163 (decimal expansion of sqrt(5)), A010686 (repeat 1, 5), A090550, A134976.

Cf. A010499 (decimal expansion of 3*sqrt(5)).

SetDefaultRealField(RealField(105)); n:=(5+3*Sqrt(5))/10; Reverse(Intseq(Floor(10^104*n))); // Arkadiusz Wesolowski, Jan 07 2018

Digits := 1000:  (5+3*sqrt(5.0))/10; # Muniru A Asiru, Jan 22 2018

RealDigits[(5 + 3 Sqrt[5])/10, 10, 1001][[1]] (* Georg Fischer, Apr 02 2020 *)

(5 + 3*sqrt(5))/10 \\ Michel Marcus, Apr 20 2020

Equals (A134976 + 8)/10. - R. J. Mathar, Apr 12 2010
From Arkadiusz Wesolowski, Jan 07 2018: (Start)
Equals A001622^2 / sqrt(5).
Equals lim_{n -> infinity} A000045(n+2) / A001622^n. (End)
Equals 1/A090550 + 1. - Michel Marcus, Apr 20 2020
Minimal polynomial is 5x^2 - 5x - 1 (this number is an algebraic number but not an algebraic integer). - Alonso del Arte, Apr 20 2020
Equals lim_{k->oo} Fibonacci(k+2)/Lucas(k). - Amiram Eldar, Feb 06 2022

A379135 Decimal expansion of the midradius of a pentakis dodecahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 17 2024

The pentakis dodecahedron is the dual polyhedron of the truncated icosahedron.

			1.4756836610416140907689600838494857255268212565695...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.
Index entries for algebraic numbers, degree 2.

Cf. A379132 (surface area), A379133 (volume), A379134 (inradius), A379136 (dihedral angle).
Cf. A205769 (midradius + 1 of a truncated icosahedron with unit edge length).
Cf. A010499.

First[RealDigits[(11 + Sqrt[45])/12, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentakisDodecahedron", "Midradius"], 10, 100]]

(11 + 3*sqrt(5))/12 \\ Charles R Greathouse IV, Feb 05 2025

Equals (11 + 3*sqrt(5))/12 = (11 + A010499)/12.

A377697 Decimal expansion of the midradius of a truncated dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 05 2024

			2.9270509831248422723068802515484571765804637697...

Eric Weisstein's World of Mathematics, Truncated Dodecahedron.
Wikipedia, Truncated dodecahedron.

Cf. A377694 (surface area), A377695 (volume), A377696 (circumradius), A377698 (Dehn invariant, negated).
Cf. A239798 (analogous for a regular dodecahedron).
Cf. A010499, A205769.

First[RealDigits[(5 + Sqrt[45])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "Midradius"], 10, 100]]

Equals (5 + 3*sqrt(5))/4 = (5 + A010499)/4.

Equals A205769 - 1/2.

A385804 Decimal expansion of the volume of a triaugmented dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 09 2025

The triaugmented dodecahedron is Johnson solid J_61.

			8.56762745781210568076720062887114294145115942427...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Triaugmented dodecahedron.

Cf. A385805 (surface area).

Cf. A010499, A102769, A179552, A377697, A385695, A385802.

First[RealDigits[5/8*(7 + Sqrt[45]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J61", "Volume"], 10, 100]]

Equals (5/8)*(7 + 3*sqrt(5)) = (5/8)*(7 + A010499).
Equals A102769 + 3*A179552.
Equals the largest root of 16*x^2 - 140*x + 25.
Equals A377697^2. - Hugo Pfoertner, Jul 13 2025

A010135 Continued fraction for sqrt(45).

Original entry on oeis.org

6, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2

Offset: 0

N. J. A. Sloane

			6.708203932499369089227521006... = 6 + 1/(1 + 1/(2 + 1/(2 + 1/(2 + ...)))). - _Harry J. Smith_, Jun 06 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A010499 (decimal expansion), A041076/A041077 (convergents).

ContinuedFraction[Sqrt[45],300] (* Vladimir Joseph Stephan Orlovsky, Mar 07 2011 *)

{ allocatemem(932245000); default(realprecision, 17000); x=contfrac(sqrt(45)); for (n=0, 20000, write("b010135.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 06 2009

G.f.: (6 + x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + 6*x^6)/(1 - x^6). - Stefano Spezia, Jul 27 2025

A041076 Numerators of continued fraction convergents to sqrt(45).

Original entry on oeis.org

6, 7, 20, 47, 114, 161, 2046, 2207, 6460, 15127, 36714, 51841, 658806, 710647, 2080100, 4870847, 11821794, 16692641, 212133486, 228826127, 669785740, 1568397607, 3806580954, 5374978561, 68306323686

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 (0,0,0,0,0,322,0,0,0,0,0,-1).

Cf. A010499, A041077.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[45],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 22 2011*)
Numerator[Convergents[Sqrt[45], 30]] (* Vincenzo Librandi, Oct 25 2013 *)

a(n) = 322*a(n-6)-a(n-12). G.f.: -(x^11-6*x^10+7*x^9-20*x^8+47*x^7-114*x^6-161*x^5-114*x^4-47*x^3-20*x^2-7*x-6)/((x^2-3*x+1)*(x^2+3*x+1)*(x^4-3*x^3+8*x^2-3*x+1)*(x^4+3*x^3+8*x^2+3*x+1)). [Colin Barker, Jul 18 2012]

A041077 Denominators of continued fraction convergents to sqrt(45).

Original entry on oeis.org

1, 1, 3, 7, 17, 24, 305, 329, 963, 2255, 5473, 7728, 98209, 105937, 310083, 726103, 1762289, 2488392, 31622993, 34111385, 99845763, 233802911, 567451585, 801254496, 10182505537, 10983760033, 32150025603, 75283811239, 182717648081, 258001459320, 3278735159921

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 (0,0,0,0,0,322,0,0,0,0,0,-1).

Cf. A010499, A041076.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[45],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 22 2011*)
Denominator[Convergents[Sqrt[45], 30]] (* Vincenzo Librandi, Oct 24 2013 *)
LinearRecurrence[{0,0,0,0,0,322,0,0,0,0,0,-1},{1,1,3,7,17,24,305,329,963,2255,5473,7728},40] (* Harvey P. Dale, Jun 11 2022 *)

a(n) = 322*a(n-6)-a(n-12). G.f.: -(x^10-x^9+3*x^8-7*x^7+17*x^6-24*x^5-17*x^4-7*x^3-3*x^2-x-1)/((x^2-3*x+1)*(x^2+3*x+1)*(x^4-3*x^3+8*x^2-3*x+1)*(x^4+3*x^3+8*x^2+3*x+1)). [Colin Barker, Jul 18 2012]

More terms from Vincenzo Librandi, Oct 24 2013

A377798 Decimal expansion of the circumradius of a truncated icosidodecahedron (great rhombicosidodecahedron) with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 08 2024

			3.802394499851293584766836714110323209303892865251...

Eric Weisstein's World of Mathematics, Great Rhombicosidodecahedron.
Wikipedia, Truncated icosidodecahedron.

Cf. A377796 (surface area), A377797 (volume), A377799 (midradius).

Cf. A010499, A344171.

First[RealDigits[Sqrt[31/4 + Sqrt[45]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosidodecahedron", "Circumradius"], 10, 100]]

Equals sqrt(31/4 + 3*sqrt(5)) = sqrt(31/4 + A010499) = sqrt(31 + A344171)/2.

cp's OEIS Frontend

A248271 Egyptian fraction representation of sqrt(45) (A010499) using a greedy function.

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A104161 G.f.: x*(1 - x + x^2)/((1-x)^2 * (1 - x - x^2)).

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A176015 Decimal expansion of (5 + 3*sqrt(5))/10.

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A379135 Decimal expansion of the midradius of a pentakis dodecahedron with unit shorter edge length.

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A377697 Decimal expansion of the midradius of a truncated dodecahedron with unit edge length.

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A385804 Decimal expansion of the volume of a triaugmented dodecahedron with unit edge.

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A010135 Continued fraction for sqrt(45).

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A104161 G.f.: x(1 - x + x^2)/((1-x)^2 (1 - x - x^2)).